$$
h^{\prime}(x)=\frac{2 h(x)}{x \ln (x)}
$$

Is $h(x)=4 \ln (x)$ a solution to the above equation?

Question

$$
h^{\prime}(x)=\frac{2 h(x)}{x \ln (x)}
$$

Is $h(x)=4 \ln (x)$ a solution to the above equation?

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ First, we need to find the derivative of $h(x) = 4 \ln(x)$ with respect to $x$. ‖ ‖ step 2 ⋮ The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$. Therefore, $h'(x) = 4 \cdot \frac{1}{x}$. ‖ ‖ step 3 ⋮ Simplify the expression for $h'(x)$ to get $h'(x) = \frac{4}{x}$. ‖ ‖ step 4 ⋮ Now, we will check if $h(x) = 4 \ln(x)$ satisfies the given differential equation $h'(x) = \frac{2h(x)}{x \ln(x)}$. ‖ ‖ step 5 ⋮ Substitute $h(x) = 4 \ln(x)$ into the right-hand side of the differential equation to get $\frac{2 \cdot 4 \ln(x)}{x \ln(x)}$. ‖ ‖ step 6 ⋮ Simplify the right-hand side to get $\frac{8 \ln(x)}{x \ln(x)} = \frac{8}{x}$. ‖ ‖ step 7 ⋮ Compare the simplified derivative $h'(x) = \frac{4}{x}$ with the result from step 6, $\frac{8}{x}$. ‖ ‖ step 8 ⋮ Since $\frac{4}{x} eq \frac{8}{x}$, the function $h(x) = 4 \ln(x)$ is not a solution to the differential equation $h'(x) = \frac{2h(x)}{x \ln(x)}$. ‖ ∻Answer∻ ⚹ No, $h(x) = 4 \ln(x)$ is not a solution to the differential equation $h'(x) = \frac{2h(x)}{x \ln(x)}$. ⚹ ∻Key Concept∻ ⚹Verifying solutions to differential equations⚹ ∻Explanation∻ ⚹To verify if a function is a solution to a differential equation, we must check if the function satisfies the equation when substituted into it. In this case, the derivative of the proposed solution does not match the form given by the differential equation, so it is not a solution.⚹◊What is the general solution to the differential equation $h'(x) = \frac{2h(x)}{x \ln(x)}$?⍭ Generate me a similar question◊

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