Podcast
Questions and Answers
What is the slope of the tangent at point P(x, y) on a curve?
What is the slope of the tangent at point P(x, y) on a curve?
dy/dx
What is the equation of the tangent at point P(x, y) on a curve?
What is the equation of the tangent at point P(x, y) on a curve?
Y - y = (dy/dx)(X - x)
What is the x-intercept of the tangent at point P(x, y) on a curve?
What is the x-intercept of the tangent at point P(x, y) on a curve?
x - y(dy/dx)
What is the equation of the normal at point P(x, y) on a curve?
What is the equation of the normal at point P(x, y) on a curve?
What is the length of the tangent at point P(x, y) on a curve?
What is the length of the tangent at point P(x, y) on a curve?
What is the length of the normal at point P(x, y) on a curve?
What is the length of the normal at point P(x, y) on a curve?
What is the differential of the area under a curve?
What is the differential of the area under a curve?
What is the radius of curvature at point P(x, y) on a curve?
What is the radius of curvature at point P(x, y) on a curve?
In polar coordinates, what is the tangent of the angle between the radius vector and the tangent line?
In polar coordinates, what is the tangent of the angle between the radius vector and the tangent line?
In polar coordinates, what is the length of the sub-tangent?
In polar coordinates, what is the length of the sub-tangent?
In polar coordinates, what is the length of the sub-normal?
In polar coordinates, what is the length of the sub-normal?
The curve in which the portion of the tangent included between the coordinate axes is bisected at the point of contact is a rectangular hyperbola.
The curve in which the portion of the tangent included between the coordinate axes is bisected at the point of contact is a rectangular hyperbola.
What is the equation of the curve for which the normal makes equal angles with the radius vector and the initial line?
What is the equation of the curve for which the normal makes equal angles with the radius vector and the initial line?
What is the shape of a reflector that reflects light coming from a fixed source into parallel rays?
What is the shape of a reflector that reflects light coming from a fixed source into parallel rays?
What is the equation of a parabola?
What is the equation of a parabola?
What is the differential equation of the orthogonal trajectory of the family of curves xy = c?
What is the differential equation of the orthogonal trajectory of the family of curves xy = c?
What is the equation of the orthogonal trajectory of the family of confocal conics x²/(α² + λ) + y²/(α² + λ) = 1?
What is the equation of the orthogonal trajectory of the family of confocal conics x²/(α² + λ) + y²/(α² + λ) = 1?
What is the equation of the family of confocal and coaxial parabolas?
What is the equation of the family of confocal and coaxial parabolas?
A system of confocal and coaxial parabolas is self-orthogonal.
A system of confocal and coaxial parabolas is self-orthogonal.
In polar coordinates, what is the equation of the orthogonal trajectory for a family of cardioids r = a (1 - cos θ)?
In polar coordinates, what is the equation of the orthogonal trajectory for a family of cardioids r = a (1 - cos θ)?
What is the equation of the orthogonal trajectory for the family of curves rn = a sin nθ?
What is the equation of the orthogonal trajectory for the family of curves rn = a sin nθ?
What is the formula for the velocity of a body moving along a curve?
What is the formula for the velocity of a body moving along a curve?
What is the formula for the acceleration of a body moving along a curve?
What is the formula for the acceleration of a body moving along a curve?
What is Hooke's Law?
What is Hooke's Law?
What is the equation of motion for a body of mass m moving under gravity with air resistance kv²?
What is the equation of motion for a body of mass m moving under gravity with air resistance kv²?
What is the formula for velocity of escape from the earth?
What is the formula for velocity of escape from the earth?
What is the equation of the free surface of water in a cylindrical tank rotating with an angular velocity ω about its axis?
What is the equation of the free surface of water in a cylindrical tank rotating with an angular velocity ω about its axis?
What is the formula for the velocity of water flowing through a small hole?
What is the formula for the velocity of water flowing through a small hole?
What is the formula for atmospheric pressure p at a height z above sea level when the temperature is constant?
What is the formula for atmospheric pressure p at a height z above sea level when the temperature is constant?
In what way is the temperature of a body changing according to Newton's law of cooling?
In what way is the temperature of a body changing according to Newton's law of cooling?
What is the formula for the quantity of heat q flowing across a slab of area a and thickness dx?
What is the formula for the quantity of heat q flowing across a slab of area a and thickness dx?
What is the formula for the rate of decay of radio-active material?
What is the formula for the rate of decay of radio-active material?
What is the formula for the current i in an R-L series circuit?
What is the formula for the current i in an R-L series circuit?
What is the formula for the rate of change of salt content (u) in a tank?
What is the formula for the rate of change of salt content (u) in a tank?
What is the formula for the time (t) it takes for the salt content to change from u = 40 to u = 80 in a tank?
What is the formula for the time (t) it takes for the salt content to change from u = 40 to u = 80 in a tank?
Which characteristic defines linear differential equations?
Which characteristic defines linear differential equations?
What form does a linear differential equation with constant coefficients take?
What form does a linear differential equation with constant coefficients take?
Which of the following methods can be used to find a particular integral?
Which of the following methods can be used to find a particular integral?
What is a property of the solutions of a linear differential equation?
What is a property of the solutions of a linear differential equation?
In the standard form of a linear differential equation, what do the functions $p_1, p_2, ..., p_n$ depend on?
In the standard form of a linear differential equation, what do the functions $p_1, p_2, ..., p_n$ depend on?
What is the significance of constant coefficients in linear differential equations?
What is the significance of constant coefficients in linear differential equations?
What type of equations do Cauchy’s and Legendre’s belong to?
What type of equations do Cauchy’s and Legendre’s belong to?
Which of the following is true about the solutions of simultaneous linear equations with constant coefficients?
Which of the following is true about the solutions of simultaneous linear equations with constant coefficients?
What is the first step in finding the complementary function for a differential equation?
What is the first step in finding the complementary function for a differential equation?
In solving for the roots of the auxiliary equation, which case represents two real and equal roots?
In solving for the roots of the auxiliary equation, which case represents two real and equal roots?
What form is used for a pair of imaginary roots in the solution?
What form is used for a pair of imaginary roots in the solution?
Which statement describes the method for obtaining a particular integral?
Which statement describes the method for obtaining a particular integral?
What is represented by the expression $A_1 e^{m_1 x} + A_2 e^{m_2 x} + ... + A_n e^{m_n x}$ in the context provided?
What is represented by the expression $A_1 e^{m_1 x} + A_2 e^{m_2 x} + ... + A_n e^{m_n x}$ in the context provided?
Which form is used when there are three real and equal roots in the solution?
Which form is used when there are three real and equal roots in the solution?
What is the significance of the symbol D in the context of differential equations?
What is the significance of the symbol D in the context of differential equations?
How is the complementary function for the case of a pair of equal imaginary roots expressed?
How is the complementary function for the case of a pair of equal imaginary roots expressed?
Which differential operator corresponds to the equation (D3 + 2D2 + D)y = x2e2x + sin2 x?
Which differential operator corresponds to the equation (D3 + 2D2 + D)y = x2e2x + sin2 x?
What is the general form of the differential equation represented by (D2 + 4D + 3)y = e^x sin x + xe^3x?
What is the general form of the differential equation represented by (D2 + 4D + 3)y = e^x sin x + xe^3x?
In the equation (D2 + 1)^2 y = x^4 + 2 sin x cos 3x, what type of solution approaches should be considered to solve it?
In the equation (D2 + 1)^2 y = x^4 + 2 sin x cos 3x, what type of solution approaches should be considered to solve it?
For the equation (D2 + 2D + 1)y = x cos x, which characteristic of the solution is indicated?
For the equation (D2 + 2D + 1)y = x cos x, which characteristic of the solution is indicated?
How does the equation 2(dy/dx) + 3y = e^x cos x relate to linear differential equations?
How does the equation 2(dy/dx) + 3y = e^x cos x relate to linear differential equations?
Which operator accurately reflects the structure of the equation (D2 + 6D + 11)y = 6 + cos 2x?
Which operator accurately reflects the structure of the equation (D2 + 6D + 11)y = 6 + cos 2x?
Which method would be suitable for solving (D3 + 5D2 + 7D)y = e^2x cosh x?
Which method would be suitable for solving (D3 + 5D2 + 7D)y = e^2x cosh x?
What does the equation (D^2 + 1)^2 y = x^4 + 2 sin x cos 3x indicate about its general solution?
What does the equation (D^2 + 1)^2 y = x^4 + 2 sin x cos 3x indicate about its general solution?
When can the equation $sin(ax + b) = x$ be applied?
When can the equation $sin(ax + b) = x$ be applied?
Under which condition does $cos(ax + b) = x imes cos(ax + b)$ hold true?
Under which condition does $cos(ax + b) = x imes cos(ax + b)$ hold true?
What is required for the equation $cos(ax + b) = cos(ax + b)$ to be used?
What is required for the equation $cos(ax + b) = cos(ax + b)$ to be used?
What conditions apply when $f'(a^2) = 0$?
What conditions apply when $f'(a^2) = 0$?
What is an example of a situation where $sin(ax + b) = x^2 imes sin(ax + b)$ is true?
What is an example of a situation where $sin(ax + b) = x^2 imes sin(ax + b)$ is true?
Which equation reflects the initial condition of the differential operator $D$ when addressing $cos(2x)$?
Which equation reflects the initial condition of the differential operator $D$ when addressing $cos(2x)$?
What must be satisfied for $sin(ax + b)$ to be equated with $x$ in a function?
What must be satisfied for $sin(ax + b)$ to be equated with $x$ in a function?
In the function $f(D) = 2$, what result would be expected if $f'(D)$ is non-zero?
In the function $f(D) = 2$, what result would be expected if $f'(D)$ is non-zero?
What is the complementary function (C.F.) of the differential equation given?
What is the complementary function (C.F.) of the differential equation given?
Which expression correctly represents the particular integral (P.I.) derived from the equation?
Which expression correctly represents the particular integral (P.I.) derived from the equation?
What method is used to find the roots of the characteristic equation D^2 - 4D + 4 = 0?
What method is used to find the roots of the characteristic equation D^2 - 4D + 4 = 0?
What is the general solution of the differential equation after combining the C.F. and P.I.?
What is the general solution of the differential equation after combining the C.F. and P.I.?
How is the particular integral evaluated in terms of integration?
How is the particular integral evaluated in terms of integration?
What form does the complete solution take when expressed as a sum of functions?
What form does the complete solution take when expressed as a sum of functions?
What does the term 'C.S.' refer to in the context of solving the differential equation?
What does the term 'C.S.' refer to in the context of solving the differential equation?
In the given context, how many constants of integration are involved in the C.F.?
In the given context, how many constants of integration are involved in the C.F.?
What is the complementary function (C.F.) of the equation $D^2 - 4y = 0$?
What is the complementary function (C.F.) of the equation $D^2 - 4y = 0$?
What is the particular integral (P.I.) for the equation $y'' - 4y = x \sinh x$?
What is the particular integral (P.I.) for the equation $y'' - 4y = x \sinh x$?
In the solution process for $D^2 - 4y = x \sinh x$, what form does the auxiliary equation take?
In the solution process for $D^2 - 4y = x \sinh x$, what form does the auxiliary equation take?
Which of the following represents the homogeneous solution for the differential equation $D^2y - 4y = 0$?
Which of the following represents the homogeneous solution for the differential equation $D^2y - 4y = 0$?
What additional term is included in the complementary solution when considering the full equation involving $sinh x$?
What additional term is included in the complementary solution when considering the full equation involving $sinh x$?
The expression $\frac{e^x - e^{-x}}{2}$ corresponds to which of the following?
The expression $\frac{e^x - e^{-x}}{2}$ corresponds to which of the following?
What can be concluded about the term $3D^2 + 2D + 2$ when applied to an equation?
What can be concluded about the term $3D^2 + 2D + 2$ when applied to an equation?
What function represents the general solution $y = c_1 e^x + c_2 e^{2x} + P.I.$ for the equation $D^2 - 4y = 0$?
What function represents the general solution $y = c_1 e^x + c_2 e^{2x} + P.I.$ for the equation $D^2 - 4y = 0$?
In solving for $D^2 - 4y = x sinh x$, what is the significance of calculating the roots of the characteristic equation?
In solving for $D^2 - 4y = x sinh x$, what is the significance of calculating the roots of the characteristic equation?
Flashcards
Slope of Tangent
Slope of Tangent
At any point on a curve, the slope of the tangent line is given by the derivative dy/dx.
Equation of Tangent
Equation of Tangent
The equation of the tangent line at point P(x, y) is given by y = (dy/dx)(X - x) + y, where X and Y are the coordinates on the tangent line.
Length of Sub-Tangent
Length of Sub-Tangent
The length of the sub-tangent is the distance along the x-axis between the point of contact and the point where the tangent intersects the x-axis. It is calculated as y(dx/dy).
Length of Sub-Normal
Length of Sub-Normal
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Polar Sub-Tangent
Polar Sub-Tangent
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Polar Sub-Normal
Polar Sub-Normal
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Orthogonal Trajectories
Orthogonal Trajectories
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Finding Orthogonal Trajectories (Cartesian)
Finding Orthogonal Trajectories (Cartesian)
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Finding Orthogonal Trajectories (Polar)
Finding Orthogonal Trajectories (Polar)
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Newton's Second Law
Newton's Second Law
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Hooke's Law
Hooke's Law
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Resisted Motion Equation (Drag force proportional to velocity)
Resisted Motion Equation (Drag force proportional to velocity)
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Resisted Motion Equation (Drag force proportional to velocity squared)
Resisted Motion Equation (Drag force proportional to velocity squared)
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Terminal Velocity
Terminal Velocity
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Escape Velocity
Escape Velocity
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Centrifugal Force
Centrifugal Force
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Equation of a Rotating Liquid Surface
Equation of a Rotating Liquid Surface
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Discharge Rate of Water Through a Hole
Discharge Rate of Water Through a Hole
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Atmospheric Pressure Equation (Constant Temperature)
Atmospheric Pressure Equation (Constant Temperature)
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Kirchhoff's Current Law (KCL)
Kirchhoff's Current Law (KCL)
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Kirchhoff's Voltage Law (KVL)
Kirchhoff's Voltage Law (KVL)
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Voltage Drop Across Resistance
Voltage Drop Across Resistance
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Voltage Drop Across Inductance
Voltage Drop Across Inductance
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Voltage Drop Across Capacitance
Voltage Drop Across Capacitance
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RL Circuit Equation (Constant EMF)
RL Circuit Equation (Constant EMF)
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RL Circuit Equation (Sinusoidal EMF)
RL Circuit Equation (Sinusoidal EMF)
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RLC Circuit Equation (Constant EMF)
RLC Circuit Equation (Constant EMF)
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Solution of the First Order Differential Equation
Solution of the First Order Differential Equation
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Initial Conditions
Initial Conditions
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Linear Differential Equation
Linear Differential Equation
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Order of a Differential Equation
Order of a Differential Equation
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Complementary Function (CF)
Complementary Function (CF)
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Particular Integral (PI)
Particular Integral (PI)
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General Solution
General Solution
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Linear Dependence of Solutions
Linear Dependence of Solutions
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Operator D
Operator D
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Inverse Operator D^-1
Inverse Operator D^-1
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Complementary Function
Complementary Function
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Particular Integral
Particular Integral
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Auxiliary Equation (A.E.)
Auxiliary Equation (A.E.)
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Roots of A.E.
Roots of A.E.
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C.F. (Complementary Function)
C.F. (Complementary Function)
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Working Procedure
Working Procedure
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Symbolic Form
Symbolic Form
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Solving the A.E. for 'D'
Solving the A.E. for 'D'
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Symbolic Form of DE
Symbolic Form of DE
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Why is P.I. Needed?
Why is P.I. Needed?
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General Solution = CF + PI
General Solution = CF + PI
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How to find a PI
How to find a PI
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Operator 'D' Inverse
Operator 'D' Inverse
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Linearly Independent Solutions
Linearly Independent Solutions
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How to find Linearly Independent Solutions
How to find Linearly Independent Solutions
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P.I. of a differential equation
P.I. of a differential equation
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How to find P.I. for cos(ax + b)
How to find P.I. for cos(ax + b)
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Conditions for P.I. of cos(ax + b)
Conditions for P.I. of cos(ax + b)
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Example: Finding P.I. of (D^3 + 1)y = cos(2x)
Example: Finding P.I. of (D^3 + 1)y = cos(2x)
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What is f(D^2) in a differential equation?
What is f(D^2) in a differential equation?
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Why use the operator 1/f(D^2) for P.I.?
Why use the operator 1/f(D^2) for P.I.?
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What if f(a^2) = 0 when finding P.I. of cos(ax + b)?
What if f(a^2) = 0 when finding P.I. of cos(ax + b)?
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What happens if f'(a^2) = 0 when finding the P.I. of cos(ax + b)?
What happens if f'(a^2) = 0 when finding the P.I. of cos(ax + b)?
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Differential Equation (DE)
Differential Equation (DE)
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Order of a DE
Order of a DE
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Linear DE
Linear DE
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Homogeneous DE
Homogeneous DE
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Non-Homogeneous DE
Non-Homogeneous DE
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What is the purpose of the complementary function (CF) in solving a non-homogeneous differential equation?
What is the purpose of the complementary function (CF) in solving a non-homogeneous differential equation?
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What is the purpose of the particular integral (PI) in solving a non-homogeneous differential equation?
What is the purpose of the particular integral (PI) in solving a non-homogeneous differential equation?
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How do you obtain the general solution of a non-homogeneous differential equation?
How do you obtain the general solution of a non-homogeneous differential equation?
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Study Notes
Applications of Differential Equations of First Order
- Differential equations of the first order are used to model practical problems.
- Geometric applications involve curves, tangents, normals, subtangents, and subnormals.
- Cartesian coordinates are used to find the slope and equations of tangents and normals.
- Polar coordinates enable calculation of polar subtangents and subnormals.
- Orthogonal trajectories are curves that intersect every member of a given family of curves at right angles.
- Physical applications, such as simple electric circuits, Newton's law of cooling, heat flow, radioactive decay, chemical reactions, resisted motion, vertical motion, and orbital motion problems are solved using first order differential equations.
Geometric Applications
-
Cartesian coordinates:
- Slope of tangent at a point (x, y) on a curve f(x, y) = 0 is dy/dx.
- Equation of tangent at (x, y) is Y - y1 = (X - x1)(dy/dx).
- X-intercept (tangent) is x - y(dx/dy).
- Y-intercept (tangent) is y - x(dy/dx).
- Equation of normal at (x, y) is Y - y1 = - (X -x1)(dx/dy).
- Length of tangent is y√(1 + (dx/dy)²).
- Length of normal is y√(1 + (dy/dx)²).
- Length of subtangent is y(dx/dy).
- Length of subnormal is y(dy/dx).
- Radius of curvature at (x, y) is p = [(1 + (dy/dx)²)³/²]/|d²y/dx²|.
- Differential of area is ydx or xdy.
-
Polar coordinates:
- ψ = θ + φ.
- tan φ = r(dθ/dr).
- p = r sin φ.
- p² = r² + (dr/dθ)²
Orthogonal Trajectories
- Two families of curves are orthogonal trajectories if each member of one family cuts each member of the other family at right angles.
- To find orthogonal trajectories of a family of curves F(x, y, c) = 0:
- Eliminate the constant c from the given equation to form its differential equation f(x, y, dy/dx) = 0.
- Replace (dy/dx) with - (dx/dy) in the differential equation.
- Solve the resulting differential equation to obtain the orthogonal trajectories.
Physical Applications
-
Simple electric circuits:
- The voltage drops across the components (resistance, inductance, and capacitance) in a circuit are related to the current and its rate of change.
- Kirchhoff's laws (voltage and current) can be used to derive differential equations describing the circuit.
-
Newton's law of cooling:
The rate of change of temperature of an object is proportional to the difference between its temperature and the surrounding temperature.
- Heat flow:
The rate of heat flow through a material is proportional to the temperature gradient.
- Radioactive decay:
The rate of decay of a radioactive material is proportional to the amount of material remaining.
- Chemical reactions:
The rate of a chemical reaction may depend on the concentrations of the reactants.
-
Resisted motion:
-
A moving body is opposed by a force per unit mass, proportional to the displacement x and the velocity squared b v².
-
Vertical motion:
- The forces acting on a falling particle are its weight mg and resistance (proportional to velocity squared) mλv upwards,
-
Orbital Motion:
-
The acceleration of a falling particle is proportional to 1/r^2 where r is the distance from the center.
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Description
Explore the practical applications of first order differential equations, covering both geometric and physical contexts. This quiz delves into Cartesian and polar coordinates, tangent and normal lines, and real-world scenarios like cooling and decay. Test your understanding of these concepts through problem-solving and application-based questions.
