Video Transcript
Is the function 𝑦 equals 𝑒 to the power of five 𝑥 minus 𝑒 to the 𝑥 a solution to the differential equation 𝑦 prime equals five 𝑦 minus four 𝑒 to the power 𝑥?
The idea here is that we check the possible solution by substituting it into the differential equation to see if we get a true statement. Our differential equation is 𝑦 prime equals five 𝑦 minus four 𝑒 to the 𝑥. And we’re going to make substitutions for 𝑦 prime and 𝑦. On the right-hand side, we can replace 𝑦 with the possible solution 𝑒 to the five 𝑥 minus 𝑒 to the 𝑥, to get five multiplied by 𝑒 to the five 𝑥 minus 𝑒 to the 𝑥 minus four 𝑒 to the 𝑥.
Let’s now distribute the parentheses. And then we can simplify this expression by collecting like terms, to get five 𝑒 to the five 𝑥 minus nine 𝑒 to the 𝑥. On the left-hand side of the differential equation, we have 𝑦 prime. Remember, we’re checking if 𝑒 to the five 𝑥 minus 𝑒 to the 𝑥 is a solution to this. So we need to find 𝑦 prime for our possible solution in order to make this substitution.
We remember the general rule that the derivative of 𝑒 raised to 𝑓 of 𝑥, a function of 𝑥, is 𝑓 prime of 𝑥 multiplied by 𝑒 to the power of 𝑓 of 𝑥. And so 𝑦 prime equals five 𝑒 to the power of five 𝑥 minus 𝑒 to the 𝑥. And so that’s the left-hand side of our differential equation.
So is this a true statement? Well, it doesn’t work for any value of 𝑥. So we can conclude that 𝑒 to the five 𝑥 minus 𝑒 to the 𝑥 is not a solution to this differential equation.
