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
