Mathematics

Problem

A student claims that the function

y(x)=x^{2}e^{3x}

is a solution of the differential equation

\dfrac{dy}{dx}=2x e^{3x}

. Which of the following best explains why the claim is incorrect? a) The derivative of

x^{2}

2x

, and treating

e^{3x}

as a constant gives exactly

2x e^{3x}

, so no discrepancy exists. b) Differentiating

y

produces an additional

3x^{2}e^{3x}

term that is not present in the right–hand side. c) Applying the product rule but omitting the chain-rule constant produces an extra

x^{2} e^{3x}

term, showing the differential equation is not satisfied. d) A correct differentiation actually gives

6x e^{3x}

because the derivative of

x^{2}

6x

, so the right–hand side of the DE is too small.

Intuitive explanation

The key is to actually differentiate $y(x) = x^2 e^{3x}$ using the product rule and see what you get. When you apply the product rule correctly—treating $x^2$ and $e^{3x}$ as separate factors and remembering that $e^{3x}$ has its own derivative from the chain rule—you'll find the actual derivative includes an extra term that doesn't match the claimed equation. The student likely forgot to account for all the pieces that come from differentiating a product.

Intuitive explanation

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