Solve differential equation 6(y-x^2)dx+xdy=0

hexacordoK 2020-11-23 Answered
Solve differential equation 6(yx2)dx+xdy=0
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Expert Answer

firmablogF
Answered 2020-11-24 Author has 92 answers

6(yx2)dx+xdy=0
(6(yx2)dx)/(xdx)+(xdy)/(xdx)=0
(6(yx2)dx)/(xdx)+(xdy)/(xdx)=0
6((yx2)/x)+dy/dx=0 dy/dx+(6/x)y=6x (1)
Now, equation (1) is first order linear differential equation
We know that solution of first order linear differential equation is given by
y.(I.F.)=(I.F.)6xdx (2)x
where, I.F.=e6/xdx
I.F.=e6dx/x (:dx/x=lnx+c)
=e6(lnx)
=elnx6
=x6
Now, solving equation(2) by using I.F.
y(x6)=x6(6x)dx
=6x7dx (:xndx=(xn+1)/(n+1)+c
=6(x8/8)+c
=(3x8)/4+c (3)
Now, using y(1)= 1 in equation (3)
1(1)6=(3(1)8)/4+c
1=3/4+c
c=13/4
c=(43)/4
c=1/4
Now, putting value of c in equation (3) yx6=(3x8)/4+1/4
Hence, solution of given differential equation is yx6=(3x8)/4+1/4

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