Second-order ODE involving two functions I am wondering how

Zoie Phillips

Zoie Phillips

Answered question

2022-03-31

Second-order ODE involving two functions
I am wondering how to find a general analytical solution to the following ODE:
dydtd2xdt2=dxdtd2ydt2
The solution method might be relatively simple; but right now I don't know how to approach this problem.

Answer & Explanation

Charlie Haley

Charlie Haley

Beginner2022-04-01Added 14 answers

Step 1
I find it more convenient to rewrite the equation using Newton's notation. Instead of writing dxdt, it is more helpful to write x'. Thus, the equation is
xy=xy.
Now, suppose x=0. Then x0 is trivial, so every differentiable function yRR satisfies the equation. This holds analogously if y=0. Otherwise, we can divide by x'y', thus
yy=xx.
Step 2
There are four cases to consider from here: x<0 and y<0;x<0 and y>0;x>0 and y<0; and x>0 and y>0. These cases simplify the equation respectively
ln(x)+C=ln(y)
ln(x)+C=ln(y)
ln(x)+C=ln(y)
ln(x)+C=ln(y)
which are equivalent to
eCx=y
eCx=y
eCx=y
eCx=y.
These cases simply reduce to y=Ax
where A0. Therefore, we have that y(t)=Ax(t)+B, where A0. Remember that this is in the case when neither x nor y is a constant function.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?