Step By Step Calculus » 16.3 - Linear Differential Equations

show_hiddenExpand All [+]
Synopsis
First-order linear differential equations can be written in the form

\displaystyle { \textbf{(First-order Linear DE) }\qquad\qquad \frac{dy}{dx}+f(x)y=g(x). }\displaystyle { \textbf{(First-order Linear DE) }\qquad\qquad \frac{dy}{dx}+f(x)y=g(x). }

One can solve such equations by using the following steps.
Step 1: (Convert) Express the equation as \displaystyle \frac{dy}{dx}+f(x)y=g(x),\displaystyle \frac{dy}{dx}+f(x)y=g(x), and identify f(x)f(x).
Step 2: (Calculate Factor) Find the integrating factor I(x)I(x) by evaluating \displaystyle I(x)=e^{\int f(x)dx}.\displaystyle I(x)=e^{\int f(x)dx}.
Step 3: (Multiply) Multiply \displaystyle \frac{dy}{dx}+f(x)y=g(x)\displaystyle \frac{dy}{dx}+f(x)y=g(x) by I(x)I(x) so the equation becomes \frac{d}{dx} (I(x)y)=I(x)g(x).\frac{d}{dx} (I(x)y)=I(x)g(x).
Step 4: (Take Antiderivative) Take antiderivative of both sides so
\displaystyle I(x)y=\int I(x)g(x)dx+c\Longleftrightarrow y=\frac{\int I(x)g(x)dx}{I(x)}+\frac{c}{I(x)}
\displaystyle I(x)y=\int I(x)g(x)dx+c\Longleftrightarrow y=\frac{\int I(x)g(x)dx}{I(x)}+\frac{c}{I(x)}
Step 5: (Solve) Solve for cc from supplied data.