A student claims that the function is a solution of the differential equation

MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations

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.