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x^3 dy/dx + 3x^2y = x y(2) = 0

Solution

x³dy/dx +3x²y=x

x³dy +3x²ydx=xdx

divide by x³ on both sides

(x³/x³)dy+3(x²/x³)ydx=(x/x³)dx

dy+(3/x)ydx=(1/x²)dx

integrating factor =e^(3/x)dx

integrating factor =e^(3lnx)

integrating factor =e^{(lnx^3)}

integrating factor =x³

multiply on both sides by x³

x³dy+(3/x)x³ydx=x³(1/x²)dx

dyx³+y3x²dx=xdx

product rule:u\'v +uv\' =(uv)\'

(yx³)\'=xdx

integrate on both sides

(yx³)\'= xdx

yx³=(1/2)x² +c

given y(2)=0

0*2³=(1/2)2² +c

0=2+c

c=-2

yx³=(1/2)x² -2

y=(1/2)(x²/x³) +(c/x³)

y=(1/(2x))+(c/x³) is the solution