Differential Equations: Direct Integration

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Questions and Answers

Which of the following methods involves manipulating an equation to eliminate derivatives?

  • Homogeneous equations
  • Direct integration (correct)
  • Variable separation
  • Differentiation

What is the primary goal when solving a differential equation?

To find the relationship between y and x

Separating variables is always the most direct method for solving any differential equation.

False (B)

When solving a first-order differential equation, the constant of integration provides the one ______ constant that we always get.

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

Match the method to the type of differential equation it is best suited for:

<p>Direct Integration = Equations that can be expressed in the form dy/dx = f(x) Separating Variables = Equations where the right-hand side can be expressed as products or quotients of functions of x or y Homogeneous Equations = Equations where all terms have the same degree</p> Signup and view all the answers

In the context of solving differential equations, what is the purpose of direct integration?

<p>To eliminate derivatives and find a direct relationship between variables (A)</p> Signup and view all the answers

What form must a differential equation take if it is to be solved by direct integration?

<p>$dy/dx = f(x)$ (D)</p> Signup and view all the answers

The constant of integration is optional when solving differential equations.

<p>False (B)</p> Signup and view all the answers

In the equation $dy/dx = 3x^2 - 6x + 5$, what mathematical operation is used to find 'y'?

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

When an equation is in the form $dy/dx = f(x, y)$, and the variable 'y' prevents direct integration, one must find another ______ of solution.

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

What characteristic of a differential equation makes the 'separating the variables' technique applicable?

<p>The ability to express the right-hand side as products or quotients of functions of x and y (B)</p> Signup and view all the answers

The technique of separating variables can only be applied to first-order differential equations.

<p>False (B)</p> Signup and view all the answers

What is the first step in solving a differential equation by separating variables if you have $(y+1) \frac{dy}{dx} = 2x$?

<p>Integrate both sides with respect to x</p> Signup and view all the answers

In the method of separating variables, the goal is to get all 'y' terms with 'dy' on one side and all ______ terms with 'dx' on the other side.

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

Match each differential equation with the appropriate first step in solving by separating variables:

<p>$\frac{dy}{dx} = (1+x)(1+y)$ = $\frac{1}{1+y} dy = (1+x) dx$ $\frac{dy}{dx} = \frac{1+y}{2+x}$ = $\frac{1}{1+y} dy = \frac{1}{2+x} dx $</p> Signup and view all the answers

When solving homogeneous equations, what substitution is typically made?

<p>$y = vx$ (D)</p> Signup and view all the answers

The substitution y = vx transforms a homogeneous equation into a non-separable form.

<p>False (B)</p> Signup and view all the answers

If you determine that a differential equation is homogeneous, what is the next step after making the substitution y = vx?

<p>Differentiate with respect to x</p> Signup and view all the answers

A homogeneous differential equation is characterized by having all terms with the same ______.

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

Match each equation to whether or not it is homogeneous:

<p>$\frac{dy}{dx} = \frac{x+3y}{2x}$ = Homogeneous $\frac{dy}{dx} = \frac{x^2+y^2}{xy}$ = Homogeneous $\frac{dy}{dx} = \frac{2xy+3y^2}{x^2+2xy}$ = Homogeneous $\frac{dy}{dx} = x^2 + y $ = Not Homogeneous</p> Signup and view all the answers

What distinguishes a linear first-order differential equation from other types?

<p>It can be written in the form $dy/dx + Py = Q$, where P and Q are functions of x. (C)</p> Signup and view all the answers

The integrating factor method is exclusively used for non-linear differential equations.

<p>False (B)</p> Signup and view all the answers

What is the term for the factor that you multiply to a linear differential equation to make it easily integrable?

<p>Integrating factor</p> Signup and view all the answers

In the linear first-order differential equation $\frac{dy}{dx} + 5y = e^{2x}$, the equation is multiplied by $e^{5x}$. This expression, $e^{5x}$, represents the ______ factor.

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

Match each component of the linear first-order differential equation with its role in finding the solution:

<p>P and Q = Functions of x used to determine the integrating factor and the final solution Integrating Factor (IF) = A factor multiplied to the equation to make it integrable y.IF = ∫Q.IF dx = Formula used to find the general solution after applying the integrating factor</p> Signup and view all the answers

What is the integrating factor (IF) for the differential equation $\frac{dy}{dx} + y = x^3$?

<p>$e^x$ (C)</p> Signup and view all the answers

After multiplying a linear first-order differential equation by its integrating factor, the left-hand side becomes the derivative of a product.

<p>True (A)</p> Signup and view all the answers

What is the general form of the integrating factor for a linear first-order differential equation?

<p>$e^{\int P dx}$</p> Signup and view all the answers

In solving the linear differential equation $\frac{dy}{dx} - y = x$, the value of $\int P dx$ is ______.

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

Match each linear differential equation with its integrating factor:

<p>$\frac{dy}{dx} + 3y = e^{4x}$ = $e^{3x}$ $x\frac{dy}{dx} + y = x \sin x$ = x $\tan x \frac{dy}{dx} + y = \sec x$ = $\sin x$</p> Signup and view all the answers

In the context of solving differential equations, what does 'degree' refer to when determining if an equation is homogeneous?

<p>The power of the variable in each term of the equation (C)</p> Signup and view all the answers

If a differential equation can be solved by direct integration, it is impossible to solve it by separating variables.

<p>False (B)</p> Signup and view all the answers

What is the purpose of finding an integrating factor when solving a linear first-order differential equation?

<p>To make the equation integrable</p> Signup and view all the answers

To solve the equation $\frac{dy}{dx} = xy - y$, factor out ______.

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

Match the characteristics to the kind of Ordinary Differential Equation (ODE) it describes:

<p>Homogeneous ODE = All terms have the same degree. Linear ODE = Can be expressed as $dy/dx + Py = Q$, where P and Q are functions of x. Separable ODE = Can be written in the form $f(y)dy = g(x)dx$.</p> Signup and view all the answers

Why is it important to express the right-hand side (RHS) of a separable differential equation in ‘x-factors’ and ‘y-factors’?

<p>To separate the variables correctly before integration (D)</p> Signup and view all the answers

Once a homogeneous equation has been transformed using the appropriate substitution, it is always directly integrable without further steps.

<p>False (B)</p> Signup and view all the answers

How does the integrating factor simplify solving a linear first-order differential equation?

<p>It makes the equation integrable</p> Signup and view all the answers

In the process of solving differential equations, the phrase 'particular solution' refers to a solution where specific ______ are satisfied.

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

What would be the integrating factor for solving $dy/dx + y = 0.5$?

<p>$e^{x}$ (D)</p> Signup and view all the answers

Flashcards

Solving a differential equation

Finding the function that satisfies a differential equation.

Manipulation Goal

Eliminate derivatives to find a relationship between variables.

Direct Integration

Solving equations by integrating both sides directly.

Form for Direct Integration

dy/dx = f(x), solved by integrating f(x).

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Separating Variables

Solving by expressing the equation as functions of x and y.

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Variable Separation Form

dy/dx = f(x) * F(y)

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Homogeneous Equation Substitution

Replace y with vx, where v is a function of x.

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Homogeneous Equation

All terms have the same total degree.

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Linear Equation Method

Multiply by integrating factor e^(∫Pdx).

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Linear Equation Form

dy/dx + Py = Q where P and Q are functions of x.

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Integrating Factor

e^(∫Pdx)

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Study Notes

  • Module Note for Engineering Mathematics 2A (EEEM216), 1st Semester 2025 concerns the solution of differential equations.
  • Recommended textbooks include:
  • "Engineering Mathematics" by K.A. Stroud (2020, 8th edition, Bloomsbury Academic, ISBN: 9781352010275)
  • "Advanced Engineering Mathematics" by K.A. Stroud (2003, 4th edition, Palgrave Macmillan).

Solving Differential Equations

  • Solving a differential equation involves finding the function for which the equation holds true
  • This requires manipulating the equation to eliminate derivates resulting in a relation between y and x.

Method 1: By Direct Integration

  • If a differential equation can be arranged in the form dy/dx = f(x), it can be solved using simple integration.
  • Constant integration must be included, providing one arbitrary constant when solving a first-order differential equation.

Example 1 of Direct Integration

  • Given dy/dx = 3x² - 6x + 5, integrating both sides results in y = x³ - 3x² + 5x + C

Example 2 of Direct Integration

  • Solve x(dy/dx) = 5x³ + 4
  • dy/dx = 5x² + 4/x
  • y = (5x³/3) + 4ln(x) + C

Example 3 of Direct Integration

  • Find the particular solution of eˣ(dy/dx) = 4, with y = 3 when x = 0.
  • Rewrite as dy/dx = 4e⁻ˣ.
  • y = ∫4e⁻ˣ dx = -4e⁻ˣ + C
  • y = -4e⁻ˣ + 7

Method 2: By Separating the Variables

  • If the equation is of the form dy/dx = f(x, y), direct integration is prevented by the variable y on the right-hand side,
  • necessitating other solution methods.

Example 1 of Separating Variables

  • Given dy/dx = 2x/(y + 1)
  • Rewrite it as (y + 1)dy/dx = 2x
  • ∫(y + 1) dy = ∫2x dx
  • results in y²/2 + y = x² + C.

Example 2 of Separating Variables

  • Given dy/dx = (1 + x)(1 + y)
  • Rewrite as (1 / (1 + y)) dy/dx = 1 + x.
  • ∫(1 / (1 + y)) dy = ∫(1 + x) dx
  • ln(1 + y) = x + (x²/2) + C.
  • F(y) dy/dx = f(x)
  • ∫F(y) dy = ∫f(x) dx.

Example 3 of Separating Variables

  • Solving dy/dx = (1 + y) / (2 + x)
  • Rewrite equation as 1/(1+y) * dy/dx = 1/(2+x)
  • integrating both sides gives ln(1 + y) = ln(2 + x) + C
  • The constant C rewritten as ln A to simplify ln(1 + y) = ln(2 + x) + ln A = ln[A(2 + x)]
  • 1 + y = A(2 + x)

Example 4 of Separating Variables

  • To solve dy/dx = (y² + xy²) / (x²y - x²),
  • express the right-hand side in 'x-factors' and 'y-factors',
  • rearrange the equation so that the ‘y-factors’ and dy are on the left-hand side (LHS) ‘x-factors’ and dx are on the right-hand side (RHS).

Example 5 of Separating Variables

  • To solve dy/dx = (y² - 1) / x, so dy = ((y² - 1) / x)*dx,
  • 1/(y² - 1) dy = - dx/x
  • integrate both sided ∫1/(y² - 1) dy = ∫1/x dx
  • solution: 1/2 ln |(y - 1)/(y + 1)| = ln x + C

Method 3: Homogeneous Equations

  • Homogeneous equations can be solved by substituting y = vx.
  • This method applies when the right-hand side (RHS) of the equation can’t be expressed as ‘x-factors’ and ‘y-factors’ for variable separation.

Example of Homogeneous Equations

  • Given dy/dx = (x + 3y) / 2x, substitute y = vx, where v is a function of x.
  • Differentiate concerning x to obtain dy/dx = v + x(dv/dx,
  • rewrite and solve separating the variables.
  • The total degree in x and y should be the same to solve.

Example 1 of Homogeneous Equations

  • To solve dy/dx = (x² + y²) / xy, substitute y = vx as the equation is homogeneous
  • The terms on the RHS have a degree of 2.
  • Given y = vx, dy/dx = v + x (dv/dx),
  • v + x dv/dx = (x² + (vx)²) / (x(vx))
  • v + x dv/dx = (1 + v²) / v
  • x dv/dx = (1 + v²) / v - v
  • x dv/dx = 1/v
  • ∫ v * dv = ∫ 1/x dx
  • results in v²/2 = ln x + C.

Example 2 of Homogeneous Equations

  • To solve dy/dx = (2xy + 3y²) / (x² + 2xy)
  • Determine that the degree of each term in the equation is the same.
  • Then make a substitution of y = vx, where v is a function of x.
  • The equation may be expressed as x(dv/dx) = (v + v^2) / (1 + 2v)

Example 3 of Homogeneous Equations

  • To solve (x² + y²) dy/dx = xy
  • put y = vx
  • rewrite to v + x *(dv/dx)= v/(1+v²)

Method 4: Linear Equations and Use of Integrating Factor

  • Linear equations in the form of dy/dx + 5y = e²ˣ

  • Multiplying both sides by an integrating factor e⁵ˣ transforms the left-hand side into the derivative of y⋅e⁵ˣ.

  • Integrating both sides gives y.e⁵ˣ = ∫e⁷ˣ dx = e⁷ˣ/7 + C

  • Equations can be in the form dy/dx + Py = Q, where P and Q are functions of x or constants.

  • Multiply both sides by an integrating factor e∫Pdx to solve an equation such as this.

Steps for Solving Linear First-Order Equations

  • Consider dy/dx + Py = Q, integrating factor is IF = e^∫Pdx)
  • multiply through dy/dx.e^∫Pdx + Py.e^∫Pdx = Q.e^∫Pdx
  • gives that the LHS is the derivative of y.e^∫Pdx
  • Integrate both sides.
  • Let the integrating factor be IF to simplify the result y.IF=∫Q.IF
  • Given dy/dx - y = x, P = -1, and the integration factor is e^-x.

Exercise Solutions for Linear Equations

  • First example final solution y = (e^(4x) / 7) + Ce^(-3x), is reached through an integrating factor of e^(3x)
  • Second example final solution xy = sinx - xcosx + C, integrating factor is x
  • Third example final solution y sinx = x + c, integrating factor is sinx

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