Final answer to the problem
$y=\left(\sum_{n=0}^{\infty } \frac{{\left(\left(-\frac{1}{2}\right)\right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(n!\right)}+\frac{1}{e^{\frac{1}{2}x^2}}+C_0\right)e^{\frac{1}{2}x^2}$
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Step-by-step Solution
Learn how to solve integration techniques problems step by step online. Solve the differential equation y^'-xy=1-x. Rewrite the differential equation using Leibniz notation. 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)=1-x. 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. So the integrating factor \mu(x) is.
Final answer to the problem
$y=\left(\sum_{n=0}^{\infty } \frac{{\left(\left(-\frac{1}{2}\right)\right)}^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(n!\right)}+\frac{1}{e^{\frac{1}{2}x^2}}+C_0\right)e^{\frac{1}{2}x^2}$
