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Is there a general proposed way of solving ODE's of the form f(y,y')=0? any ideas?

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- Thread starter AdrianZ
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- #1

- 319

- 0

Is there a general proposed way of solving ODE's of the form f(y,y')=0? any ideas?

- #2

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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.

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