ODE of the form y''=f(y,y')

Solution Techniques for Problems of the Form y''=f(y,y')

A second-order differential equation of the form

can be converted into a first-order differential equation by assuming y that is
the independent variable and y' is the new dependent variable. Suppose that
we can write

Then

Substituting these expressions into the original differential equation we obtain

Dividing both sides by v(y), we obtain

This is a first-order ode for v(y). Suppose we can compute v(y). Then

This is a separable first-order differential equation for y(t).

Example

Consider the ode

This ode is of the correct type, since y'' depends does not depend
explicitly on t. Making the substitution suggested above, we convert
the ode into

This is a separable ode in the variable v. The solution is

where C_1 is a constant. This leads to the new differential equation

This is a separable ode. We have

where C_2 is a second constant. Integrating both sides, we obtain

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