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Answer 1 · 2017-01-09T19:29:15

#y = e^(1/2(x^2-1)sin^2(x^2))#

This differential equation is separable. So

#(dy)/y=x^3cos(x^2)dx#. Integrating we have

#log y=1/2(cos(x^2) + x^2 sin(x^2))+C# or

#y = C_1e^(1/2(cos(x^2) + x^2 sin(x^2)))#

The initial condition dictates

#1 = C_1e^(1/2)# so the final solution is

#y = e^(1/2(x^2-1)sin^2(x^2))#