Mathematics

Problem

Consider the differential equation

\frac{dy}{dx}=\frac{3y}{2x},\qquad x>0,

and the student work shown below attempting to verify that

y=k x^{3/2}

is a solution. Student work (with one error): 1. Claim:

\dfrac{dy}{dx}=\dfrac{3k}{2}x^{1/2}

. 2. Substitute into the differential equation:

\dfrac{3k}{2}x^{1/2}=\frac{3(k x^{3/2})}{2x}=\frac{3k}{2}x^{1/2}.

3. Conclude that the function satisfies the differential equation for all

x>0

. (a) Identify the specific step in the student work that fails to justify the conclusion and clearly describe the error. (b) Provide a complete and correct verification that

y=k x^{3/2}

satisfies the differential equation for

x>0

. (c) Explain why the condition

x>0

is essential for the verification in part (b).

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