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please explain in detailed steps the general solution of the following differential equation. given equation: y"-3y'/x=x solution: C1x^4/4-x^3/3+C2 required : detailed steps in the solution of the above

please explain in detailed steps the general solution of the following differential equation.                                                    given equation:   y"-3y'/x=x                                                     solution:           C1x^4/4-x^3/3+C2                                           required :        detailed steps in the solution of the above

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2 Answers

SAGAR SINGH - IIT DELHI
879 Points
10 years ago

Dear student,

Let y=e^mx

dy/dx=me^mx

2y/dx2=m2e^mx

Now substitute this and solve the equation......

vikas askiitian expert
509 Points
10 years ago

y11 - 3y1/x = x

d/dx (y1) - 3y1/x = x             ..................1                    

put y1 = z theneq 1 becomes

dz/dx - 3z/x = x               ..............2

now this is a linear differential equation withintegration factor = 1/x3

z/x3 = x/x3 dx

z/x3 = (1/x2)dx = -1/x + c1

z = -x2 + c1x3 ...................3

now , dy/dx = z so

dy/dx = -x2 + c1x3

dy = (-x2 + c1x3)dx

 y  = -x3/3 + c1x4/4 + c2           .................5

this is the required equation

 

approve if u like my ans

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