Maths with Lemon

Integrating Factors

IB Mathematics: Analysis and Approaches Higher Level · Lesson 10.4 · AHL 5.18

Learn how to solve first-order linear differential equations by recognising a product-rule derivative.

Lesson 10.4 worksheet (PDF) · Previous lesson: Separable and Homogeneous Differential Equations

1. When Should You Use an Integrating Factor?

What you need to know

  • The equation must be linear in \(y\) and its first derivative.
  • Rearrange it into standard form before identifying \(P(x)\) and \(Q(x)\).
  • The integrating factor converts the left side into one product-rule derivative.
  • Use any initial condition only after obtaining the general solution.

Standard form

\[\boxed{\frac{dy}{dx}+P(x)y=Q(x)}\]

The integrating factor is

\[\boxed{\mu(x)=e^{\int P(x)\,dx}}\]

Multiplying the equation by \(\mu(x)\) gives

\[\boxed{\frac{d}{dx}\left(\mu(x)y\right)=\mu(x)Q(x)}\]

2. Why Does the Method Work?

The product rule connection

  • Recall the product rule for \(\mu(x)y\).
  • Choose \(\mu\) so that its derivative is \(P(x)\mu\).
  • Then integrate the whole product derivative in one step.

Derivation

\[\frac{d}{dx}(\mu y)=\mu\frac{dy}{dx}+\mu^{\prime}y\]
\[\mu^{\prime}=P(x)\mu\]
\[\mu y=\int \mu(x)Q(x)\,dx+C\]
\[\boxed{y=\frac{1}{\mu(x)}\left(\int\mu(x)Q(x)\,dx+C\right)}\]

3. Worked Example from Your Worksheet

Solution strategy

  • Identify \(P(x)=2/x\).
  • Find the integrating factor.
  • Multiply through and simplify using the product rule.
  • Integrate and isolate \(y\).

Solve for \(x>0\)

Solve \(\frac{dy}{dx}+\frac{2y}{x}=x^3\)
\[\mu(x)=e^{\int 2/x\,dx}=x^2\]
\[x^2\frac{dy}{dx}+2xy=x^5\]
\[\frac{d}{dx}(x^2y)=x^5\]
\[x^2y=\frac{x^6}{6}+C\]
\[\boxed{y=\frac{x^4}{6}+\frac{C}{x^2}}\]

Substitute into the original equation to check your result.

4. Exponential and Trigonometric Examples

Spot the integrating factor

  • For \(y^{\prime}+ay=Q(x)\), use \(e^{ax}\).
  • For \(P(x)=1/x\), use \(|x|\) on an interval excluding zero; \(x\) is convenient for \(x>0\).
  • For \(P(x)=\tan x\), an integrating factor is \(\sec x\) on intervals where \(\cos x\ne0\).

Quick worked examples

Example A: \(y^{\prime}+y=e^x\)
\[\mu=e^x,\qquad (e^xy)^{\prime}=e^{2x}\]
\[\boxed{y=\frac12e^x+Ce^{-x}}\]
Example B: \(y^{\prime}+y\tan x=2\sin x\)
\[\mu=\sec x\]
\[(y\sec x)^{\prime}=2\tan x\]
\[\boxed{y=\cos x\left(-2\ln|\cos x|+C\right)}\]

5. Particular Solutions and Initial Conditions

Exam method

  • Find the general solution first.
  • Substitute the given point \((x_0,y_0)\).
  • Solve for the constant \(C\).
  • State the interval of validity if the equation has a restricted domain.

Worksheet-style example

Solve \(y^{\prime}+\frac{2y}{x}=12x\), given \(y(1)=5\), \(x>0\)
\[(x^2y)^{\prime}=12x^3\]
\[x^2y=3x^4+C\]
\[y=3x^2+\frac{C}{x^2}\]

Using \(y(1)=5\), we obtain \(C=2\).

\[\boxed{y=3x^2+\frac{2}{x^2}}\]

6. Application: Falling Through a Liquid

Interpret the model

  • Velocity changes due to a constant driving term and a resistive term.
  • A terminal velocity occurs when the acceleration becomes zero.
  • An integrating factor gives the full velocity function, not only the terminal value.

Worksheet model

Solve \(\frac{dv}{dt}=10-2v\), with \(v(0)=v_0\)
\[\frac{dv}{dt}+2v=10\]
\[(e^{2t}v)^{\prime}=10e^{2t}\]
\[\boxed{v(t)=5+(v_0-5)e^{-2t}}\]

As \(t\to\infty\), the exponential term tends to zero.

\[\boxed{v_{\mathrm{terminal}}=5}\]

7. Interactive Integrating Factor Explorer

Explore a linear equation

  • Change the coefficient \(a\), the forcing \(b\), and the initial value.
  • See the exact solution of \(y^{\prime}+ay=b\).
  • Observe how the solution approaches the equilibrium \(b/a\) when \(a>0\).

Model: \(y^{\prime}+ay=b\)

8. HL Practice and Revision

From your Lesson 10.4 worksheet

  • Solve \(y^{\prime}-y=e^x\).
  • Solve \(y^{\prime}+y/x=x^2\).
  • Investigate integrating factors involving \(\tan x\).
  • Apply substitutions such as \(z=y^2\) or \(z=\sin y\) to obtain linear equations.
  • Work through the extension problems involving falling motion and second-order equations.

Common mistakes

  • Forgetting to divide by the coefficient of \(y^{\prime}\) before finding \(P(x)\).
  • Missing the constant of integration.
  • Confusing \(\mu y\) with \(\mu\,dy/dx\).
  • Dropping domain restrictions in logarithmic or trigonometric integrating factors.

Open the full worksheet

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