Second-order ODE involving two functions I am wondering how to find

adiadas8o7 2022-04-30 Answered
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.
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Answers (1)

Felicity Carter
Answered 2022-05-01 Author has 16 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.
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