Solutions to Differential Equations - At A Glance

Answer

If y = e^x then y ' = e^x and y" = e^x. The left-hand side of the d.e. comes out to be

y' + y" = e^x + e^x = 2e^x

and the right-hand side of the d.e. comes out to be

2y = 2(e^x) = 2e^x.

Since the left-hand side and right-hand side of the d.e. came out the same, y = e^x is a solution to this differential equation.

Answer

If y = xe^x then

y' = xe^x + e^x.

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

xy = xe^x.

Since the left- and right- hand sides of the d.e. are not equal when y = xe^x, this is NOT a solution to the differential equation.

Answer

When P = e^-t, the left-hand side of the d.e. is

The right-hand side of the d.e. is

P(1 - P) = P - P²,

which is not equal to -P. Since the left- and right- hand sides of the d.e. are not equal, P = e^-t is not a solution to the differential equation.

Answer

When y = x², we get

Then the left-hand side of the d.e. is

The right-hand side of the d.e. is

Since the left- and right- hand sides of the d.e. are equal, y = x² IS a solution to this differential equation.

Answer

When y = xⁿ, we have y' = nx^{n – 1} and

y" = n(n – 1)x^{(n – 2)}.

Then the left-hand side of the d.e. is

x²y " + ny = x²[n(n – 1)x^{(n – 2)}] + n[xⁿ]

= n(n – 1)xⁿ + nxⁿ

= n²xⁿ – nxⁿ + nxⁿ

= n²xⁿ.

The right-hand side of the d.e. is

n²y = n²xⁿ.

Since the left- and right- hand sides of the d.e. are the same, y = xⁿ is a solution to this differential equation.

We may also encounter problems that look something like this:

Show that the function such-and-such is a solution to the d.e. blah-blah."

While the word "show" may strike terror in your heart, don't give in to fear! Problems that say "show" are actually easier than ones that look like this:

"Determine whether the function such-and-such is a solution to the d.e. blah-blah."

Regardless of whether a problem says "show" or "determine", do the same thing: evaluate the left-hand side of the d.e., evaluate the right-hand side of the d.e., and see if you get the same thing from both sides.

Here's the difference between these types of problems: If a problem says "determine," you need to figure out whether the function is a solution to the d.e. or not. If a problem says "show," it's telling you the answer you'll get if you do the work right.

Answer

We know that we should get the same thing when we evaluate the two sides of the differential equation. If y = 5x² + 2 then

y' = 10x and y" = 10. The left-hand side of the d.e. is

and the right-hand side of the d.e. is

2y = 2(5x² + 2) = 10x² + 4.

Since we got the same thing on both sides of the d.e., we have shown that y = 5x² + 2 is a solution to this differential equation.

Answer

We know that we should get different things for the two sides of the differential equation. If y = 5x² then y' = 10x and y" = 10. The left-hand side of the d.e. is

and the right-hand side of the d.e. is

2y = 2(5x²) = 10x².

Since we got different things for the two sides of the d.e., we have shown that y = 5x² is not a solution to this differential equation.

Answer

We know that we should get the same thing for both sides of the differential equation. Since the right-hand side is zero, this means we should get zero for the left-hand side. Let's see if that happens.

If f (x) = sin x then f '(x) = cos x and f ⁽²⁾(x) = -sin x. The left-hand side of the d.e. is

f ⁽²⁾(x) + f (x) = (-sin x) + (sin x) = 0.

Since the two sides of the d.e. are equal when f (x) = sin x, we have shown that f (x) = sin x is a solution to this differential equation.

Answer

We know that we should get the same thing for both sides of the differential equation.

If f (x) = e^xsin x then

f '(x) = e^xsin x + e^xcos x

and

f "(x) = e^xsin x + e^xcos x + e^xcos x – e^xsin x = 2e^xcos x.

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

The left-hand side of the differential equation is

2f '(x) – 2f (x) = 2(e^xsin x + e^xcos x) – 2e^xsin x = 2e^xcos x.

Since we got the same thing for both sides of the d.e., we've shown that f (x) = e^xsin x is a solution to this differential equation.

Answer

We know that we should get the same thing for both sides of the differential equation.If y = ln x then . The left-hand side of the d.e. is

e^y = e^{ln x} = x

and the right-hand side is

Since we got the same thing for both sides of the d.e., we've shown that y = ln x is a solution to this differential equation.