Applications of First Order Differential Equations
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Questions and Answers

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?

Y - y = (dy/dx)(X - x)

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?

<p>Y - y = (-dx/dy)(X - x)</p> Signup and view all the answers

What is the length of the tangent at point P(x, y) on a curve?

<p>y√(1 + (dx/dy)²)</p> Signup and view all the answers

What is the length of the normal at point P(x, y) on a curve?

<p>y√(1 + (dy/dx)²)</p> Signup and view all the answers

What is the differential of the area under a curve?

<p>ydx or xdy</p> Signup and view all the answers

What is the radius of curvature at point P(x, y) on a curve?

<p>(1 + (dy/dx)²)³/² / (d²y/dx²)</p> Signup and view all the answers

In polar coordinates, what is the tangent of the angle between the radius vector and the tangent line?

<p>rdθ/dr</p> Signup and view all the answers

In polar coordinates, what is the length of the sub-tangent?

<p>r²dθ/dr</p> Signup and view all the answers

In polar coordinates, what is the length of the sub-normal?

<p>dr/dθ</p> Signup and view all the answers

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.

<p>True</p> Signup and view all the answers

What is the equation of the curve for which the normal makes equal angles with the radius vector and the initial line?

<p>r = a(1 - cos θ)</p> Signup and view all the answers

What is the shape of a reflector that reflects light coming from a fixed source into parallel rays?

<p>a parabola</p> Signup and view all the answers

What is the equation of a parabola?

<p>y² = 2cx + c²</p> Signup and view all the answers

What is the differential equation of the orthogonal trajectory of the family of curves xy = c?

<p>x² - y² = c'</p> Signup and view all the answers

What is the equation of the orthogonal trajectory of the family of confocal conics x²/(α² + λ) + y²/(α² + λ) = 1?

<p>x² + y² = 2a² log x + c</p> Signup and view all the answers

What is the equation of the family of confocal and coaxial parabolas?

<p>y² = 4a(x + a)</p> Signup and view all the answers

A system of confocal and coaxial parabolas is self-orthogonal.

<p>True</p> Signup and view all the answers

In polar coordinates, what is the equation of the orthogonal trajectory for a family of cardioids r = a (1 - cos θ)?

<p>r = a'(1 + cos θ)</p> Signup and view all the answers

What is the equation of the orthogonal trajectory for the family of curves rn = a sin nθ?

<p>rn = b cos nθ</p> Signup and view all the answers

What is the formula for the velocity of a body moving along a curve?

<p>v = ds/dt</p> Signup and view all the answers

What is the formula for the acceleration of a body moving along a curve?

<p>a = d²s/dt²</p> Signup and view all the answers

What is Hooke's Law?

<p>T = λε/l or T = ke</p> Signup and view all the answers

What is the equation of motion for a body of mass m moving under gravity with air resistance kv²?

<p>mv(dv/dx) = mg - kv²</p> Signup and view all the answers

What is the formula for velocity of escape from the earth?

<p>v = √(2gR)</p> Signup and view all the answers

What is the equation of the free surface of water in a cylindrical tank rotating with an angular velocity ω about its axis?

<p>y = (ω²/2g)x² + h</p> Signup and view all the answers

What is the formula for the velocity of water flowing through a small hole?

<p>v = 0.6√(2gy)</p> Signup and view all the answers

What is the formula for atmospheric pressure p at a height z above sea level when the temperature is constant?

<p>p = poe-gz/k</p> Signup and view all the answers

In what way is the temperature of a body changing according to Newton's law of cooling?

<p>The rate of change of temperature is proportional to the difference between the temperature of the body and the temperature of the surrounding medium.</p> Signup and view all the answers

What is the formula for the quantity of heat q flowing across a slab of area a and thickness dx?

<p>q = − kadT/dx</p> Signup and view all the answers

What is the formula for the rate of decay of radio-active material?

<p>du/dt = -ku</p> Signup and view all the answers

What is the formula for the current i in an R-L series circuit?

<p>i = (E/R)(1 - e-Rt/L)</p> Signup and view all the answers

What is the formula for the rate of change of salt content (u) in a tank?

<p>du/dt = 4 − 2u/50</p> Signup and view all the answers

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?

<p>t = 25 log(100/(100-u))</p> Signup and view all the answers

Which characteristic defines linear differential equations?

<p>The dependent variable and its derivatives occur only in the first degree.</p> Signup and view all the answers

What form does a linear differential equation with constant coefficients take?

<p>It is expressed in terms of derivatives and constants.</p> Signup and view all the answers

Which of the following methods can be used to find a particular integral?

<p>Method of variation of parameters</p> Signup and view all the answers

What is a property of the solutions of a linear differential equation?

<p>Any linear combination of solutions is also a solution.</p> Signup and view all the answers

In the standard form of a linear differential equation, what do the functions $p_1, p_2, ..., p_n$ depend on?

<p>Only the independent variable.</p> Signup and view all the answers

What is the significance of constant coefficients in linear differential equations?

<p>They simplify the nature of the solutions.</p> Signup and view all the answers

What type of equations do Cauchy’s and Legendre’s belong to?

<p>Linear differential equations</p> Signup and view all the answers

Which of the following is true about the solutions of simultaneous linear equations with constant coefficients?

<p>They may have infinitely many solutions.</p> Signup and view all the answers

What is the first step in finding the complementary function for a differential equation?

<p>Writing the auxiliary equation and solving for D</p> Signup and view all the answers

In solving for the roots of the auxiliary equation, which case represents two real and equal roots?

<p>(c1 + c2 x)em1 x + c3 em3 x + ...</p> Signup and view all the answers

What form is used for a pair of imaginary roots in the solution?

<p>eax(c1 cos βx + c2 sin βx)</p> Signup and view all the answers

Which statement describes the method for obtaining a particular integral?

<p>Can be used generally for any given case</p> Signup and view all the answers

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?

<p>The complementary function</p> Signup and view all the answers

Which form is used when there are three real and equal roots in the solution?

<p>(c1 + c2 x + c3 x^2) em1 x + c4 em4 x + ...</p> Signup and view all the answers

What is the significance of the symbol D in the context of differential equations?

<p>It represents a differential operator</p> Signup and view all the answers

How is the complementary function for the case of a pair of equal imaginary roots expressed?

<p>eax[(c1 + c2x) cos βx + (c3 + c4x) sin βx]</p> Signup and view all the answers

Which differential operator corresponds to the equation (D3 + 2D2 + D)y = x2e2x + sin2 x?

<p>Third Order</p> Signup and view all the answers

What is the general form of the differential equation represented by (D2 + 4D + 3)y = e^x sin x + xe^3x?

<p>Second Order Non-Homogeneous</p> Signup and view all the answers

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?

<p>Method of Variation of Parameters</p> Signup and view all the answers

For the equation (D2 + 2D + 1)y = x cos x, which characteristic of the solution is indicated?

<p>Repeated Roots in Characteristic Equation</p> Signup and view all the answers

How does the equation 2(dy/dx) + 3y = e^x cos x relate to linear differential equations?

<p>It is a linear first-order differential equation.</p> Signup and view all the answers

Which operator accurately reflects the structure of the equation (D2 + 6D + 11)y = 6 + cos 2x?

<p>Second Order Non-Homogeneous</p> Signup and view all the answers

Which method would be suitable for solving (D3 + 5D2 + 7D)y = e^2x cosh x?

<p>Exact Solution via Laplace Transforms</p> Signup and view all the answers

What does the equation (D^2 + 1)^2 y = x^4 + 2 sin x cos 3x indicate about its general solution?

<p>It will have complex conjugate roots contributing oscillatory components.</p> Signup and view all the answers

When can the equation $sin(ax + b) = x$ be applied?

<p>When $f'(a^2) eq 0$</p> Signup and view all the answers

Under which condition does $cos(ax + b) = x imes cos(ax + b)$ hold true?

<p>When $f'(a^2) = 0$</p> Signup and view all the answers

What is required for the equation $cos(ax + b) = cos(ax + b)$ to be used?

<p>If $f'(a^2) eq 0$</p> Signup and view all the answers

What conditions apply when $f'(a^2) = 0$?

<p>The function must switch to considering higher order derivatives</p> Signup and view all the answers

What is an example of a situation where $sin(ax + b) = x^2 imes sin(ax + b)$ is true?

<p>When $f'(a^2) = 0$ and $f''(a^2) eq 0$</p> Signup and view all the answers

Which equation reflects the initial condition of the differential operator $D$ when addressing $cos(2x)$?

<p>$(D^3 + 1)y = cos(2x)$</p> Signup and view all the answers

What must be satisfied for $sin(ax + b)$ to be equated with $x$ in a function?

<p>f'(a^2) must not equal zero</p> Signup and view all the answers

In the function $f(D) = 2$, what result would be expected if $f'(D)$ is non-zero?

<p>The function behaves normally</p> Signup and view all the answers

What is the complementary function (C.F.) of the differential equation given?

<p>(c1 + c2x) e2x</p> Signup and view all the answers

Which expression correctly represents the particular integral (P.I.) derived from the equation?

<p>8e2x (x2 sin 2x)</p> Signup and view all the answers

What method is used to find the roots of the characteristic equation D^2 - 4D + 4 = 0?

<p>Factoring</p> Signup and view all the answers

What is the general solution of the differential equation after combining the C.F. and P.I.?

<p>y = (c1 + c2x) e2x + 8e2x (x2 sin 2x)</p> Signup and view all the answers

How is the particular integral evaluated in terms of integration?

<p>Using the integral involving sin 2x</p> Signup and view all the answers

What form does the complete solution take when expressed as a sum of functions?

<p>C.F. + P.I.</p> Signup and view all the answers

What does the term 'C.S.' refer to in the context of solving the differential equation?

<p>Complete Solution</p> Signup and view all the answers

In the given context, how many constants of integration are involved in the C.F.?

<p>Two constants</p> Signup and view all the answers

What is the complementary function (C.F.) of the equation $D^2 - 4y = 0$?

<p>c1e^2x + c2e^{-2x}</p> Signup and view all the answers

What is the particular integral (P.I.) for the equation $y'' - 4y = x \sinh x$?

<p>$\frac{x}{2} (e^x - e^{-x})$</p> Signup and view all the answers

In the solution process for $D^2 - 4y = x \sinh x$, what form does the auxiliary equation take?

<p>$D^2 - 4 = 0$</p> Signup and view all the answers

Which of the following represents the homogeneous solution for the differential equation $D^2y - 4y = 0$?

<p>c1e^{2x} + c2e^{-2x}</p> Signup and view all the answers

What additional term is included in the complementary solution when considering the full equation involving $sinh x$?

<p>1/2 x sinh x</p> Signup and view all the answers

The expression $\frac{e^x - e^{-x}}{2}$ corresponds to which of the following?

<p>sinh x</p> Signup and view all the answers

What can be concluded about the term $3D^2 + 2D + 2$ when applied to an equation?

<p>It is a second-order polynomial with constant coefficients.</p> Signup and view all the answers

What function represents the general solution $y = c_1 e^x + c_2 e^{2x} + P.I.$ for the equation $D^2 - 4y = 0$?

<p>y = c_1 e^x + c_2 e^{2x} + \frac{x}{2} (e^x - e^{-x})$</p> Signup and view all the answers

In solving for $D^2 - 4y = x sinh x$, what is the significance of calculating the roots of the characteristic equation?

<p>It determines the structure of the complementary function.</p> Signup and view all the answers

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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Related Documents

Chapter 12 Applications of Differential Equations of First Order PDF
Higher Engineering Mathematics - Chapter 13 PDF

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.

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