Final answer to the problem
$e^{\frac{1}{2}x^2}y=2\sum_{n=0}^{\infty } \frac{\left(\frac{1}{2}\right)^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(n!\right)}+C_0$
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Step-by-step Solution
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- Exact Differential Equation
- Linear Differential Equation
- Separable Differential Equation
- Homogeneous Differential Equation
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
- Integrate by substitution
- Integrate by parts
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1
Rewrite the differential equation using Leibniz notation
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$x\frac{dy}{dx}+x^2y=2x$
Learn how to solve problems step by step online. Solve the differential equation xy^'+x^2y=2x. Rewrite the differential equation using Leibniz notation. Divide all the terms of the differential equation by x. Simplifying. 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)=2. In order to solve the differential equation, the first step is to find the integrating factor \mu(x).
Final answer to the problem
$e^{\frac{1}{2}x^2}y=2\sum_{n=0}^{\infty } \frac{\left(\frac{1}{2}\right)^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(n!\right)}+C_0$
