Solutions to Differential Equations Examples

Determine if x = 5 is a solution to the equation

x² + 6 = 4x + 11.

We evaluate the left-hand side of the equation at x = 5:

(5)² + 6 = 31.

Then we evaluate the right-hand side of the equation at x = 5:

4(5) + 11 = 31.

Since we get the same result on both sides of the equation, x = 5 is a solution to the equation.

Determine if x = 4 is a solution to the equation

x² + 6 = 4x + 11.

We evaluate the left-hand side of the equation at x = 4:

(4)² + 6 = 22.

Then we evaluate the right-hand side of the equation at x = 4:

4(4) + 11 = 27.

Since we don't get the same result from both sides of the equation, x = 4 is not a solution to the equation.

In differential equations the variables stand for functions instead of numbers.
In the differential equation

y' = xy,

the symbols y and y' stand for functions. Therefore a solution to a differential equation is a function rather than a number.

As a Twinkie satisfies a sweet tooth, a solution to a differential equation is a function that satisfies that d.e. (in other words, makes the d.e. true).

To determine if a function is a solution to a given differential equation, we do the same thing as before: evaluate the left-hand side of the d.e., evaluate the right-hand side of the d.e., and see if we get the same thing on each side.

Determine whether the function y = x² is a solution to the d.e.

xy' = 2y.

If y = x² then y ' = 2x. Therefore the left-hand side of the differential equation is

xy' = x(2x) = 2x².

The right-hand side of the differential equation is

2y = 2(x²) = 2x².

Since we found the same thing on both sides, y = x² is indeed a solution to this differential equation.

Determine whether the function y = x² + 4 satisfies the d.e.

xy' = 2y.

If y = x² + 4 then y ' = 2x. This means the left-hand side of the differential equation is

xy' = x(2x) = 2x².

The right-hand side of the differential equation is

2y = 2(x² + 4) = 2x² + 8.

Since we did not get the same thing on both sides of the equation, y = x² + 4 does not satisfy this differential equation.

Be Careful: Work with the left-hand side of the differential equation and the right-hand side of the differential equation separately. Simplify each side, then compare them after the simplification is done.

Show that the function y = xe^x is a solution to the d.e.

We do the same work we would do if the problem said this:

Determine whether the function y = e^x is a solution to the d.e.

Since the problem says to "show that the function y = e^x is a solution," we know that we should get the same thing when we work out the left- and right-hand sides of the differential equation. Let's do it.

If y = xe^x then y' = xe^x + e^x and

y" = xe^x + e^x + e^x = xe^x + 2e^x.

This is the left-hand side of the differential equation. The right-hand side is

We got the same thing from the left- and right-hand sides of the d.e., so we have shown that y = xe^x is a solution to the differential equation.

Show that y = x ln x is not a solution to the d.e.

We need to work out the left-hand side and right-hand side of the differential equation. The problem says to show that y" is not a solution,"
so we know we should get different expressions from the left- and right-hand sides of the differential equation. Here goes.

If y = x ln x then

and

This is the left-hand side of the differential equation. The right-hand side is

Since we got different expressions from the left- and right-hand sides of the d.e., we have shown that y = x ln x is not a solution to the differential equation.