Solve $3y' + 6xy = 6e^{-x}$ where $y(0)=1$.

integration ordinary-differential-equations

Based on the reformulation of the given problem, we can proceed as follows in order to solve it: \begin{align*} 3y' + 6xy = 6e^{-x^{2}} & \Longleftrightarrow y' + 2xy = 2e^{-x^{2}}\\\\ & \Longleftrightarrow e^{x^{2}}y' + 2xe^{x^{2}}y = 2\\\\ & \Longleftrightarrow (e^{x^{2}}y)' = 2\\\\ & \Longleftrightarrow e^{x^{2}}y = 2x +c\\\\ & \Longleftrightarrow y(x) = 2xe^{-x^{2}} + ce^{-x^{2}} \end{align*}

Hopefully this helps !

Related

relation between roots and coefficient in a cubic polynomial
Is Whyburn's theorem on irreducible maps optimal?
What are the values of $a$ and $b$ so that $\Bbb Z_2\times\Bbb Z_3\times\Bbb Z_4\times\Bbb Z_9$ is isomorphic with $\Bbb Z_a\times\Bbb Z_b$?
A question involving Lindeberg-Levy CLT
Show that $nY_n$ converges in distribution
Is matrix $AA^T + tI$ always regular? [closed]
Prove that $f(x):=\sqrt{x}$ on $(0,+\infty)$ is $n$-times differentiable.
A $\sigma$-algebra that is complete as a Boolean algebra?
Left Kan Extension along identity?
Are subnormal and polar subnormal equal?
Open subsets in a manifold as submanifold of the same dimension?
Prove that $f$ is integrable if and only if $\sum^\infty_{n=1} \mu(\{x \in X : f(x) \ge n\}) < \infty$