Final answer to the problem
$e^{\frac{2}{25}x^2}y=\frac{2}{5}\sum_{n=0}^{\infty } \frac{\left(\frac{2}{25}\right)^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(n!\right)}+C_0$
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Step-by-step Solution
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- Exact Differential Equation
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1
Calculate the power $5^2$
$25\left(\frac{dy}{dx}\right)+4xy=10$
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$25\left(\frac{dy}{dx}\right)+4xy=10$
Learn how to solve problems step by step online. Solve the differential equation 5^2dy/dx+4xy=10. Calculate the power 5^2. Divide all the terms of the differential equation by 25. 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)=\frac{4x}{25} and Q(x)=\frac{2}{5}. In order to solve the differential equation, the first step is to find the integrating factor \mu(x).
Final answer to the problem
$e^{\frac{2}{25}x^2}y=\frac{2}{5}\sum_{n=0}^{\infty } \frac{\left(\frac{2}{25}\right)^nx^{\left(2n+1\right)}}{\left(2n+1\right)\left(n!\right)}+C_0$
