First-order differential equations

What you’ll learn

• Solve linear equations with an integrating factor.
• Use substitutions and initial conditions.

Understand the idea

Arrange a linear first-order equation as $y'+P(x)y=Q(x)$. An integrating factor turns the left side into the derivative of a product, so a single integration gives the general solution. The integration constant is fixed only when an initial or boundary condition is supplied.

Some non-linear equations become linear after a stated substitution. Differentiate the substitution carefully and rewrite every occurrence of the original variable. Keep the domain of the integrating factor and any division restrictions visible, especially when $P(x)$ contains $1/x$.

Key formulae

$$\frac{dy}{dx}+P(x)y=Q(x)$$

$$\mu(x)=e^{\int P(x)\,dx}$$

$$\frac{d}{dx}(\mu y)=\mu Q$$

$$y=\frac1\mu\left(\int\mu Q\,dx+C\right)$$

Worked example

Solve $y'+2y=6$, given $y(0)=1$.

1. The integrating factor is $e^{2x}$.
2. Thus $(e^{2x}y)'=6e^{2x}$ and $e^{2x}y=3e^{2x}+C$.
3. So $y=3+Ce^{-2x}$; the initial condition gives $1=3+C$.

Answer: $y=3-2e^{-2x}$.