Exercise
$y'=xy+5$
Step-by-step Solution
Learn how to solve problems step by step online. Solve the differential equation y^'=xy+5. Rewrite the differential equation using Leibniz notation. Rearrange the differential equation. We can identify that the differential equation has the form: \frac{dy}{dx} + P(x)\cdot y(x) = Q(x), so we can classify it as a linear first order differential equation, where P(x)=-x and Q(x)=5. In order to solve the differential equation, the first step is to find the integrating factor \mu(x). To find \mu(x), we first need to calculate \int P(x)dx.
Solve the differential equation y^'=xy+5
Final answer to the exercise
$y=\left(5\sum_{n=0}^{\infty } \frac{{\left(\left(-\frac{1}{2}\right)\right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(n!\right)}+C_0\right)e^{\frac{1}{2}x^2}$