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Answer 1 · 2017-01-09T19:29:15

$y = e^{\frac{1}{2} (x^{2} - 1) {sin}^{2} (x^{2})}$ $y = e^(1/2(x^2-1)sin^2(x^2))$

This differential equation is separable. So

$\frac{d y}{y} = x^{3} cos (x^{2}) d x$ $(dy)/y=x^3cos(x^2)dx$ . Integrating we have

$log y = \frac{1}{2} (cos (x^{2}) + x^{2} sin (x^{2})) + C$ $log y=1/2(cos(x^2) + x^2 sin(x^2))+C$ or

$y = C_{1} e^{\frac{1}{2} (cos (x^{2}) + x^{2} sin (x^{2}))}$ $y = C_1e^(1/2(cos(x^2) + x^2 sin(x^2)))$

The initial condition dictates

$1 = C_{1} e^{\frac{1}{2}}$ $1 = C_1e^(1/2)$ so the final solution is

$y = e^{\frac{1}{2} (x^{2} - 1) {sin}^{2} (x^{2})}$ $y = e^(1/2(x^2-1)sin^2(x^2))$