Solve the given differential equation. $$ x^{4}+\left(x^{5}-11\right) y^{\prime}=0 $$ $$ y=\square $$

Question

UpStudy AI · Accepted Answer

To solve the differential equation:

$$
x^{4} + \left(x^{5} - 11\right) \frac{dy}{dx} = 0
$$

First, let's rewrite the equation to isolate $$ \frac{dy}{dx} $$:

$$
\left(x^{5} - 11\right) \frac{dy}{dx} = -x^{4}
$$

$$
\frac{dy}{dx} = \frac{-x^{4}}{x^{5} - 11}
$$

This is a separable differential equation. We can write it as:

$$
dy = \frac{-x^{4}}{x^{5} - 11} dx
$$

To integrate both sides, use the substitution $$ u = x^{5} - 11 $$. Then, the derivative $$ du $$ is:

$$
du = 5x^{4} dx \quad \Rightarrow \quad x^{4} dx = \frac{du}{5}
$$

Substituting into the integral:

$$
y = \int \frac{-x^{4}}{x^{5} - 11} dx = \int \frac{-1}{5u} du = -\frac{1}{5} \ln |u| + C
$$

Substitute back $$ u = x^{5} - 11 $$:

$$
y = -\frac{1}{5} \ln |x^{5} - 11| + C
$$

Where $$ C $$ is the constant of integration.

**Final Solution:**

$$
y = -\frac{1}{5} \ln(x^{5} - 11) + C
$$
$$ y = -\frac{1}{5} \ln(x^{5} - 11) + C $$

Solve the given differential equation.