Solve differential equation y'+3x^2y= \sin(x)e^{-x^3}, \ y(0)=1

Joni Kenny 2021-02-21 Answered
Solve differential equationy+3x2y=sin(x)ex3, y(0)=1
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Expert Answer

au4gsf
Answered 2021-02-22 Author has 95 answers
y+p(x)y=q(x), p(x)=3x2, q(x)=sin(x)ex3
epdx=e3x2dx
epdx=e3x33
epdx=ex3
That is the I.F. is μ(x)=ex3
Multiply the integrating factor on both sides of the given equation
ex3y=sin(x)ex3ex3
ex3y=sin(x)[ex3ex3=ex3+x3=1]
Integrate both sides of the above equation
ddx(ex3y)dx=sin(x)dx
ex3y=cos(x)+c
y=cos(x)+cex3
Now compute the value of constant C by applying the given condition as follows
y(0)=cos(0)+ce(0)3
1=(1)+c1 [cos(0)=1]
1=1+c
1+1=c
2=c
Now substitute the value of C for the value of y and simplify further
y=cos(x)+2ex3
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