Differential EquationsDifferential EquationsCollege

Linear First-Order Equations

Use an integrating factor.

A first-order linear differential equation has the form $\frac{dy}{dx} + P(x)y = Q(x)$ . Unlike separable equations, the variables cannot generally be split, but an integrating factor $\mu(x) = e^{\int P(x)\,dx}$ makes the left side the derivative of a product, $\frac{d}{dx}(\mu y)$ .

After multiplying through by $\mu$ , integrate both sides and divide by $\mu$ . These equations describe mixing tanks, RC circuits and objects falling with air resistance, where a rate of change depends linearly on the current amount plus an outside input.

Key terms

Integrating factor: Multiplying by it turns the left side into a derivative of a product.
$\mu(x) = e^{\int P(x)\,dx}$

Key ideas

Write the equation in standard form, with coefficient 1 on $\frac{dy}{dx}$ , before finding $\mu$ .
$\mu = e^{\int P\,dx}$ ; no constant is needed in this integral.
After multiplying, the left side is always $(\mu y)'$ .
Remember $+C$ when integrating the right side, then divide everything by $\mu$ .

Worked example

Solve $\frac{dy}{dx} + 2y = 4$ with $y(0) = 0$ .

$P = 2$ , so $\mu = e^{2x}$ .
Multiply: $(e^{2x}y)' = 4e^{2x}$ .
Integrate: $e^{2x}y = 2e^{2x} + C$ , so $y = 2 + Ce^{-2x}$ .
$y(0) = 0$ gives $C = -2$ : $y = 2 - 2e^{-2x}$ .

Answer: $y = 2 - 2e^{-2x}$

Common mistakes

Forgetting to divide by the leading coefficient first.
Multiplying only the left side by $\mu$ .
Dividing only the integral term, not the constant, by $\mu$ .

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.For $y = 2 - 2e^{-2x}$ , what value does $y$ approach as $x \to \infty$ ?
Show answer
$2$
2.Find the integrating factor of $y' + 3y = x$ , evaluated at $x = 1$ , to two decimals.
Show answer
$e^3 \approx 20.09$
3.For $y = 2 - 2e^{-2x}$ , find $y(1)$ to three decimals.
Show answer
$1.729$

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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