Is the function 𝑦 equals 𝑥 to the fourth power over four a solution to the differential equation 𝑦 prime equals 𝑥 cubed?
Recall that a differential equation is an equation which relates some function with its derivatives. The differential equation in this question is 𝑦 prime equals 𝑥 cubed. Recall that 𝑦 prime is just another way of saying d𝑦 by d𝑥, the first derivative of 𝑦 with respect to 𝑥. And we’ve been asked to check whether this function, 𝑦 equals 𝑥 to the fourth power over four, is a solution to this differential equation.
To do this, we’re going to need to make a substitution using this function into a differential equation. A possible solution is 𝑦 equals 𝑥 to the fourth power over four, which we could equivalently write as one over four multiplied by 𝑥 to the fourth power. And we can differentiate this to find 𝑦 prime. We firstly recall the power rule, which tells us that the derivative with respect to 𝑥 of 𝑥 to the 𝑛th power is equal to 𝑛 multiplied by 𝑥 to the power of 𝑛 minus one. And using this, we find that 𝑦 prime equals one over four multiplied by four 𝑥 cubed. But we can actually simplify this to 𝑥 cubed.
So, using the possible solution, we find that the left-hand side is 𝑥 cubed. And we see that the left-hand side of the differential equation and the right-hand side of the differential equation agree. Therefore, we can conclude that this is a solution to our differential equation.