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⬅ Back to Differential Equations Topics

Linear First-Order Equations

The standard form is $\frac{dy}{dx}+P(x)y=Q(x)$. Multiply by the integrating factor to convert the left side into a derivative.

$$\text{I.F.}=e^{\int P(x)\,dx}$$

Cooling-Style Linear Equation

Solve $\frac{dy}{dx}+3y=12$ if $y(0)=2$.

The integrating factor is $e^{3x}$.

$$\frac{d}{dx}\left(ye^{3x}\right)=12e^{3x}$$
$$ye^{3x}=4e^{3x}+C \quad \Rightarrow \quad y=4+Ce^{-3x}$$

Using $y(0)=2$, $C=-2$.

Final answer: $y=4-2e^{-3x}$.

Problem: Linear Equation with Constant Coefficient

Solve dy/dx + 2y = 6.

$$y_h=Ce^{-2x},\qquad y_p=3$$

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

Problem: Linear Initial-Value Problem

Solve dy/dx + y = ex, with y(0) = 2.

The integrating factor is $e^x$.

$$(e^xy)'=e^{2x} \Rightarrow e^xy=\frac{e^{2x}}{2}+C$$
$$y=\frac{e^x}{2}+Ce^{-x},\qquad 2=\frac12+C \Rightarrow C=\frac32$$

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

Problem: Convert to Linear Standard Form

Write x dy/dx + 3y = x2 in standard linear form for x not equal to zero.

$$\frac{dy}{dx}+\frac{3}{x}y=x$$

Answer: The linear coefficient is $P(x)=3/x$ and the right side is $Q(x)=x$.

Problem: Tank Mixing Linear Model

A tank model satisfies dS/dt + 0.1S = 4. Find the steady-state amount S.

At steady state, dS/dt = 0.

$$0.1S=4 \Rightarrow S=40$$

Answer: The steady-state amount is 40 units.

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