Video: Determining If a Function Is a Solution to a Differential Equation

Is the function 𝑦 = π‘₯⁴/4 a solution to the differential equation 𝑦′ = π‘₯Β³?

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Video Transcript

Is the function 𝑦 equals π‘₯ to the fourth power over four a solution to the differential equation 𝑦 prime equals π‘₯ 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 π‘₯ cubed. Recall that 𝑦 prime is just another way of saying d𝑦 by dπ‘₯, the first derivative of 𝑦 with respect to π‘₯. And we’ve been asked to check whether this function, 𝑦 equals π‘₯ 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 π‘₯ to the fourth power over four, which we could equivalently write as one over four multiplied by π‘₯ 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 π‘₯ of π‘₯ to the 𝑛th power is equal to 𝑛 multiplied by π‘₯ to the power of 𝑛 minus one. And using this, we find that 𝑦 prime equals one over four multiplied by four π‘₯ cubed. But we can actually simplify this to π‘₯ cubed.

So, using the possible solution, we find that the left-hand side is π‘₯ 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.

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