How to solve a differential equation by changing variables

Sometimes the right side of a differential equation is easier to work with as one expression. We can give that expression a new name, find an equation for it, and solve that equation first. Let’s try it with a right side that mixes both variables.

Choose a substitution that simplifies the equation

y' = 2 x + y

Let $u = 2 x + y$ . Here $u = y'$ , and differentiating $u = 2 x + y$ gives $u' = 2 + y'$ . Substitute $u$ for $y'$ to get an equation we can separate.

\frac{d u}{d x} = u + 2

Separate and integrate

First take a solution for which $u + 2$ is nonzero. Divide by $u + 2$ and integrate.

\begin{array}{l} \frac{1}{u + 2} d u = d x \\ \ln | u + 2 | = x + C_{0} \\ u + 2 = C e^{x} \end{array}

The logarithm gives an absolute value, so $C$ can be positive or negative. The equation for $u$ also has the constant solution $u = - 2$ . We can include that solution by letting $C = 0$ .

Return to the original variable

Since $u = 2 x + y$ , substitute it back and isolate $y$ .

\begin{array}{l} 2 x + y + 2 = C e^{x} \\ y = C e^{x} - 2 x - 2 \end{array}

To check the result, differentiate to get $y' = C e^{x} - 2$ . Substituting our solution into $2 x + y$ gives the same expression. With $C = 0$ , the original equation still holds and the solution is $y = - 2 x - 2$ .

When can we use the same substitution?

For $y' = a y + b x + c$ with constant coefficients and $a \neq 0$ , let $u = a y + b x + c$ . Then $u' = a u + b$ , which separates. For a different right side, check the equation for $u$ before trying to separate it.

Learn mathKrista KingMay 1, 2018variable, separable differential equations

How to solve a differential equation by changing variables

Choose a substitution that simplifies the equation

Separate and integrate

Return to the original variable

When can we use the same substitution?

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