Cauchy-Euler Diff. Eqn

  • I
  • Thread starter Arman777
  • Start date
  • #1
2,118
179
Cauchy-Euler is a type of diff equation which is described by

$$a_0x^2(\frac {d^2y} {dx^2})+a_1x(\frac {dy} {dx})+a_2y=F(x)$$

The transformation of ##x=e^t## can solve the equation.

Now, in here I didnt understand how to transform ##\frac {dy} {dx}## to ##\frac {dy} {dt}##.

it goes like this ##\frac {dy} {dx}=\frac {dy} {dt} \frac {1} {x}## and then I am stuck I should take another derivative but I couldnt do it somehow.
 
Last edited:

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
17,316
7,163
Let ##dy/dt = f## and apply the product rule for derivatives when differentiating ##f/x## wrt x.
 
  • #3
2,118
179
Let ##dy/dt = f## and apply the product rule for derivatives when differentiating ##f/x## wrt x.
Okay hmm

##\frac {dy} {dx}=\frac {f} {x}##
##\frac {d^2y} {dx^2}=\frac {df} {dx} \frac {1} {x}-\frac {f} {x^2}##

now I understand until here, but I didnt understand this term ##\frac {df} {dx}=\frac {d^2y} {dt^2}\frac {1} {x}## which its done in the book.
 
  • #4
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
17,316
7,163
What is ##df/dx## in terms of ##df/dt##?
 
  • #5
2,118
179
What is ##df/dx## in terms of ##df/dt##?
##df/dx=(df/dt)(dt/dx)##

I understand it now, thanks :angel:
 

Related Threads on Cauchy-Euler Diff. Eqn

  • Last Post
Replies
7
Views
4K
  • Last Post
Replies
9
Views
3K
  • Last Post
Replies
5
Views
6K
  • Last Post
Replies
4
Views
778
Replies
3
Views
2K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
11
Views
6K
Replies
11
Views
3K
Replies
2
Views
6K
Replies
0
Views
4K
Top