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

Bergen 2020-12-24 Answered
Reduce to first order and solve:
x2y5xy+9y=0  y1=x3
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Brighton
Answered 2020-12-25 Author has 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|
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more