Is there a general proposed way of solving ODE's of the form f(y,y')=0? any ideas?
Solve the equation f(a,t)=0 for t, considering "a" as a parameter. The result is one ore several functions t=g(a)
Let t=y' and a=y
y'=g(y)
For each function g :
dy/dx = g(y)
integrate (1/g(y))dy = dx
the result is on the forme x = f(y)
calculate x = reciprocal function of f(y)
Example :
y²+y'²-1 = 0
equation to be solved : f(a,t) = a²+t²-1 = 0
t = sqrt(1-a²)
dy/dx = sqrt(1-y²)
dx = dy/sqrt(1-y²)
x = arcsin(y) +C
y = sin(x-C)
However, some difficulties might be encountered :
- If analytical solving of equation f(a,t)=0 is not possible.
- if a primitive of the function 1/g(x) is not known
- if the reciprocal of function x=f(y) cannot be analytically computed.