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melomori [17]
3 years ago
8

Solve y' + 2xy = 2x^3, y(0) = 1

Mathematics
1 answer:
nlexa [21]3 years ago
6 0
Multiply both sides by e^{x^2}, then you get

e^{x^2}+2xe^{x^2}y=2x^3e^{x^2}
\left(e^{x^2}y\right)'=2x^3e^{x^2}
e^{x^2}y=\displaystyle2\int x^3e^{x^2}\,\mathrm dx
e^{x^2}y=e^{x^2}(x^2-1)+C
y=x^2-1+Ce^{-x^2}

Given that y(0)=1, you have

1=-1+Ce^0\implies C=2

so that the particular solution is

y=x^2-1+2e^{-x^2}
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