1.5
Guessing the Form
One powerful method for solving differential equations is an enhanced version of guess
and check: we can guess the form of the solution, and then solve for any missing
constants. For example, we might guess that a differential equation has a solution of
the form
y e ax
for some unknown value of a. We then check this solution by plugging it into the
differential equation, and then try to figure out which values of a will make the solution
work.
EXAMPLE 1
Suppose we wish to find a solution the equation
y 00 7y 0 − 10y.
We might guess that this equation has solutions of the form y e ax for some constant a. In
this case, we have
y 0 ae ax
and
y 00 a 2 e ax .
Plugging these into the equation gives
a 2 e ax 7ae ax − 10e ax
which simplifies to
a 2 e ax (7a − 10) e ax .
Now, how can we arrange it so that the left and right sides of this equation are the same? Well,
they will be the same as long as
a 2 7a − 10
The solutions to this quadratic equation are a 2 and a 5, and therefore y e 2x and
y e 5x are two solutions to the given differential equation.
EXAMPLE 2
Find a solution to the equation
x 2 y 00 2x y 0 + 10y
of the form y x a .
SOLUTION
If y x a , then
y 0 ax a−1
and
y 00 a ( a − 1) x a−2
Plugging these into differential equation gives
x 2 a ( a − 1) x a−2 2x ax a−1 + 10x a
which simplifies to
a ( a − 1) x a (2a + 10) x a .
Now, how can we arrange it so that the left and right sides of this equation are the same? Well,
they will be the same as long as
a ( a − 1) 2a + 10
Solving gives a 5 or a −2, so y x 5 and y x −2 are two solutions to this equation.
GUESSING THE FORM
2
A Closer Look Forms with Two Constants
Sometimes it works well to guess a form that involves two constants. For example, consider
the equation
y 0 y 00 2y
We might guess that this equation has solutions of the form y ax b , where a and b are
constants. The derivatives of this form are
y 0 abx b−1
and
y 00 ab ( b − 1) x b−2 .
Plugging these into the differential equation and simplifying yields
a 2 b 2 ( b − 1) x 2b−3 2ax b .
The only way for the left and right sides of this equation to be the same is if the coefficients are
the same and the exponents are the same. This gives us the following two equations:
a 2 b 2 ( b − 1) 2a
and
2b − 3 b.
The second equation tells us that b 3. Plugging this into the first equation and solving for a
yields a 0 or a 1/9. Therefore, y 0 and y 19 x 3 are two solutions to the given equation.
EXERCISES
1–4
Find all solutions to the given differential equation of the specified form.
1. y 00 3y 0 + 4y, y e ax
2. x 2 y 00 − 7x y 0 + 12y 0, y x a
3. y 0 y 00 −36y 3 , y x a
4. 5y y 00 y 0
2
+ 36y 2 , y e ax