Find the general solution of the given differential equation 4x^2y' = y'+9xe^-y

To find the general solution of the differential equation

\[
4x^2y' = y' + 9xe^{-y},
\]

we can proceed by rearranging and solving step by step.

### Step 1: Rearrange the Equation

First, let's rewrite the equation:

\[
4x^2y' - y' = 9xe^{-y}.
\]

Factor out \(y'\) on the left side:

\[
y'(4x^2 - 1) = 9xe^{-y}.
\]

### Step 2: Solve for \(y'\)

\[
y' = \frac{9xe^{-y}}{4x^2 - 1}.
\]

### Step 3: Check if the Equation is Separable

A differential equation is separable if we can write it in the form:

\[
\frac{dy}{dx} = g(x)h(y).
\]

Let's try to separate the variables \(x\) and \(y\). Rearrange the equation:

\[
\frac{dy}{dx} = \frac{9xe^{-y}}{4x^2 - 1}.
\]

Multiply both sides by \(e^y\) to get:

\[
e^y \frac{dy}{dx} = \frac{9x}{4x^2 - 1}.
\]

Now, we have:

\[
e^y dy = \frac{9x dx}{4x^2 - 1}.
\]

### Step 4: Integrate Both Sides

Integrate the left side with respect to \(y\) and the right side with respect to \(x\):

\[
\int e^y dy = \int \frac{9x dx}{4x^2 - 1}.
\]

#### Left Side:

\[
\int e^y dy = e^y.
\]

#### Right Side:

For the right side, let's perform a substitution. Let \(u = 4x^2 - 1\), so \(du = 8x dx\). The integral becomes:

\[
\int \frac{9x dx}{4x^2 - 1} = \frac{9}{8} \int \frac{du}{u} = \frac{9}{8} \ln|u| = \frac{9}{8} \ln|4x^2 - 1|.
\]

### Step 5: Combine and Solve for the General Solution

Now we have:

\[
e^y = \frac{9}{8} \ln|4x^2 - 1| + C,
\]

where \(C\) is an integration constant.

To express the solution in a more familiar form, solve for \(y\):

\[
y = \ln\left(\frac{9}{8} \ln|4x^2 - 1| + C\right).
\]

### General Solution:

\[
y = \ln\left(\frac{9}{8} \ln|4x^2 - 1| + C\right).
\]

Would you like further details or explanations on any of the steps?

Here are 5 related questions to explore:

1. How do you determine if a differential equation is separable?
2. What is the importance of the integration constant \(C\) in the general solution?
3. Can you solve a differential equation that is not separable by using other methods?
4. What would the solution look like if \(C = 0\)?
5. How does substitution help in solving integrals in differential equations?

**Tip:** When solving differential equations, always check if the equation can be simplified or rearranged to make integration easier.

Find the general solution of the given differential equation 4x^2y' = y' + 9xe^-y

Explore the general solution of the differential equation 4x^2y' = y' + 9xe^-y with step-by-step instructions. This guide covers separable equations, integration by substitution, and logarithmic integration, making it ideal for university-level calculus students.

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