Question

Math

Posted 3 months ago

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

Sign in to unlock the answer

Answer from Sia

Posted 3 months ago

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 $\frac{4}{x} \neq \frac{8}{x}$, 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.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question