Reduction of Order for Problems of the Form y''=f(t,y')
A second-order differential equation of the form
can be converted into a first-order differential equation. The procedure is called reduction of order.
Let
Using the fact that z'(t)=y''(t), we convert the original ode into the new ode
This is a first-order ode for z(t). We can use appropriate methods from first-order ode theory to compute the solution. Suppose we can compute z(t). Then y'(t)=z(t). By the Fundamental Theorem of Calculus
where C is a constant.
Example
Consider the ode
This ode is of the correct type, since y'' depends does not depend explicitly on y. Making the substitution suggested above, we convert the ode into
This is a separable ode in the variable z. The solution is
where C_1 is a constant. We then have
where C_2 is another constant.