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{\Large\bf Reduction of Order for Problems of the Form y''=f(t,y')}

\vspace{.3in}

A second-order differential equation of the form
$$
y''=f(t,y')
$$
can be converted into a first-order differential equation. The
procedure is called reduction of order.

Let 

$$
z(t)=y'(t)
$$

Using the fact that z'(t)=y''(t), we convert the original ode
into the new ode

$$
z'=f(t,z)
$$

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

$$
y(t)=\int z(t)dt+C
$$

where C is a constant.

\vspace{.3in}

{\bf Example}

\vspace{.2in}

Consider the ode

$$
y''=(y')^2
$$

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

$$
z'=z^2
$$

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

$$
z=\frac{1}{C_1-t}
$$

where C_1 is a constant. We then have

$$
y(t)=\int \frac{1}{C_1-t}dt+C_2=-\log|C_1-t|+C_2
$$

where C_2 is another constant.

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