Video: Identifying a Solution to a Differential Equation

Is the function 𝑦 = 3𝑒^(π‘₯) βˆ’ π‘₯ + 1 a solution to the differential equation 𝑦′ = π‘₯ + 𝑦?

02:26

Video Transcript

Is the function 𝑦 equals three 𝑒 to the π‘₯ minus π‘₯ add one a solution to the differential equation 𝑦 prime equals π‘₯ add 𝑦?

Our differential equation is 𝑦 prime equals π‘₯ add 𝑦. Remember that a differential equation is an equation with a function and one or more of its derivatives. In this equation, we have 𝑦 prime. But remember that this is just another way of saying d𝑦 by dπ‘₯.

To assess whether 𝑦 equals three 𝑒 to the π‘₯ minus π‘₯ add one is a solution to this differential equation. We need to substitute it into the left-hand side of our differential equation and the right-hand side of our differential equation. And we’ll then see whether the left-hand side agrees with the right-hand side.

Starting with the left-hand side, if we begin with our function 𝑦 equals three 𝑒 to the π‘₯ minus π‘₯ add one, we need to differentiate this in order to find 𝑦 prime. As we said earlier, 𝑦 prime is just another way of saying d𝑦 by dπ‘₯. So we need to differentiate our function 𝑦 with respect to π‘₯.

Starting with three 𝑒 to the π‘₯, we recall that the derivative with respect to π‘₯ of 𝑒 to the π‘₯ power is 𝑒 to the π‘₯ power. So three 𝑒 to the π‘₯ differentiates to give us three 𝑒 to the π‘₯. Then moving on to the next term, we know that π‘₯ differentiates to one. So negative π‘₯ differentiates to negative one.

And finally, we recall that constants differentiate to zero. So we find that 𝑦 prime equals three 𝑒 to the π‘₯ minus one. So that’s the left-hand side of our differential equation when we use 𝑦 equals three 𝑒 to the π‘₯ minus π‘₯ add one as a solution.

Now let’s look at the right-hand side of our differential equation. Replacing 𝑦 with our function three 𝑒 to the π‘₯ minus π‘₯ add one, we see that π‘₯ add 𝑦 is π‘₯ add three 𝑒 to the π‘₯ minus π‘₯ add one. We then see that the π‘₯s cancel. And we’re left with three 𝑒 to the π‘₯ add one.

For 𝑦 equals three 𝑒 to the π‘₯ minus π‘₯ add one to be a solution to our differential equation, we must have that the left-hand side and the right-hand side agree. On the left-hand side, we have three 𝑒 to the π‘₯ minus one. But on the right-hand side, we have three 𝑒 to the π‘₯ add one.

So, in fact, we see that the left-hand side and the right-hand side of this differential equation don’t agree. So we can conclude that this function 𝑦 is not a solution to this differential equation.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.