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Question

The solution of dydx+yx=x2y6 is

A
1x2y5=52x2+c
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B
1x5y5=5x2+c
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C
1x5y5=52x2+c
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D
1x2y2=52x2+c
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Solution

The correct option is B 1x2y5=52x2+c
This is a first order Bernocilli ODE of type, y1+p(x)y=q(x)yn
p(x)=1/x
q(x)=x2
n=6
Rearrange the ODE as,
1y6(dydx)+1y5x=x2
Perform the substitution
v=y5
dvdx=5y6(dydx)
Therefore, ODE is 15dvdx+vx=x2
dvdx(5x)v=5x2
Apply the integrating factor,
e(5/x)dx=e5sinx=eln(x5)=1/x5
1x5dvdx1x5.(5x)v=1x5.5x2
d(1/x5v)dx=5/x3
Integrating both sides,
1x5v=52x2+C
V=52x3+Cx5
1y5=52x3+Cx5=x3(5+2Cx2)2
y5=2x3(5+2Cx2)
y=(2x3(5+2Cx2))1/5
Simplified we get,
1x2y5=52x2+C

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