Question Video: Identifying a Solution to a Differential Equation Mathematics • Higher Education

Is the function 𝑦 = 3𝑒^(𝑥) − 𝑥 + 1 a solution to the differential equation 𝑦′ = 𝑥 + 𝑦?

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Video Transcript

Is the function 𝑦 equals three 𝑒 to the 𝑥 minus 𝑥 add one a solution to the differential equation 𝑦 prime equals 𝑥 add 𝑦?

Our differential equation is 𝑦 prime equals 𝑥 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 d𝑥.

To assess whether 𝑦 equals three 𝑒 to the 𝑥 minus 𝑥 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 𝑥 minus 𝑥 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 d𝑥. So we need to differentiate our function 𝑦 with respect to 𝑥.

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

And finally, we recall that constants differentiate to zero. So we find that 𝑦 prime equals three 𝑒 to the 𝑥 minus one. So that’s the left-hand side of our differential equation when we use 𝑦 equals three 𝑒 to the 𝑥 minus 𝑥 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 𝑥 minus 𝑥 add one, we see that 𝑥 add 𝑦 is 𝑥 add three 𝑒 to the 𝑥 minus 𝑥 add one. We then see that the 𝑥s cancel. And we’re left with three 𝑒 to the 𝑥 add one.

For 𝑦 equals three 𝑒 to the 𝑥 minus 𝑥 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 𝑥 minus one. But on the right-hand side, we have three 𝑒 to the 𝑥 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.