Reduce to first order and solve: x^2y''-5xy'+9y=0 y_1=x^3

Bergen

Bergen

Answered question

2020-12-24

Reduce to first order and solve:
x2y5xy+9y=0  y1=x3

Answer & Explanation

Brighton

Brighton

Skilled2020-12-25Added 103 answers

The differential equation is x2y5xy+9y=0 and one solution is y1=x3
Now, let the new solution be y=vy1=vx3 where v is again a function of x.
Then,
y=vx3
y=3vx2+x3v
y=3(2vx+x2v)+x3v+3x2v
=6vx+3x2v+x3v+3x2v
=6vx+6x2v+x3v
Substitute the values of y,y and y in x2y5xy+9y=0 and obtain the differential equation in terms of v.
x2(6vx+6x2v+x3v)5x(3vx2+x3v)+9vx3=0
6vx3+6x4v+x5v15x3v5x4v+9vx3=0
x5v+x4v=0
Now let w=v.Then the equation becomes x5w+x4w=0
Solve the equation x5w+x4w=0
x5dwdx+x4w=0
x5dwdx=x4w
dww=1xdx
dww=1xdx
ln|w|=ln|x|+c
eln|w|=eln|x|+c
w=c1x1
v=c1x1
Integrate v=c1x1 and obtain the value of v.
v=c1x1
=c1ln|x|+c2
Therefore, the general solution is y=c1x3lnx+c2x3
The second solution is y2=c1x3ln|x|

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