How Do You Solve Autonomous Second Order ODEs Like y'' = f(y)?

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SUMMARY

The discussion focuses on solving autonomous second-order ordinary differential equations (ODEs) of the form y'' = f(y). The key insight provided by Mathador is the substitution of the first derivative, defining y' as p(y), which transforms the equation into a more manageable form. This leads to the integral equation p^2 = c + ∫f(y)dy, followed by the integration of dx = dy/p(y). This method is crucial for deriving solutions to such ODEs effectively.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with integration techniques
  • Knowledge of variable substitution methods in calculus
  • Basic concepts of autonomous systems in differential equations
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  • Study the method of substitution in solving differential equations
  • Explore the implications of autonomous systems in ODEs
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Mathematicians, physics students, and engineers dealing with differential equations, particularly those interested in advanced methods for solving second-order ODEs.

muzialis
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Hi All,

I was looking for the general solution of an equation as y(x)'' = f (y), and found the attached document on the web.

It presents the solution in a way which I am not sure I understand. I tried to look at the trivila example y'' = - y, solution y = sin (x), but I am struggling in obtaining this result with the provided solution.

If anybody could help, I would be the most obliged.

All the Best

Muzialis
 

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Hi,

The trick is to define a new variable, y'=dy/dx=p(y(x)).
Now y''=dp/dy*y'=dp/dy*p=f(y).
Integrating now yields p^2=c+int(f(y))dy. The next step is to integrate dx=dy/p(y).

Mathador
 

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