Solve the initial value problem. $$ \frac{d y}{d x}=x^{4}\left(x^{5}-1\right)^{5}, y(1)=5 $$

Question

UpStudy AI · Accepted Answer

We start with the differential equation:
$$
\frac{dy}{dx} = x^4 \left(x^5 - 1\right)^5
$$
with the initial condition:
$$
y(1) = 5.
$$

**Step 1. Integrate both sides.**

We integrate with respect to $$x$$:
$$
\int \frac{dy}{dx}\,dx = \int x^4 \left(x^5 - 1\right)^5\,dx.
$$
This gives:
$$
y = \int x^4 \left(x^5 - 1\right)^5\,dx + C,
$$
where $$C$$ is the constant of integration.

**Step 2. Use substitution to evaluate the integral.**

Let
$$
u = x^5 - 1.
$$
Then, the derivative of $$u$$ with respect to $$x$$ is:
$$
\frac{du}{dx} = 5x^4 \quad \Longrightarrow \quad du = 5x^4\,dx.
$$
Solving for $$x^4\,dx$$, we have:
$$
x^4\,dx = \frac{du}{5}.
$$

Substitute into the integral:
$$
\int x^4 \left(x^5 - 1\right)^5\,dx = \int u^5 \cdot \frac{du}{5} = \frac{1}{5} \int u^5\,du.
$$

**Step 3. Evaluate the $$u$$ integral.**

Integrate $$u^5$$:
$$
\int u^5\,du = \frac{u^6}{6}.
$$
Thus:
$$
\frac{1}{5} \int u^5\,du = \frac{1}{5} \cdot \frac{u^6}{6} = \frac{u^6}{30}.
$$

Replace $$u$$ with $$x^5 - 1$$:
$$
\int x^4 \left(x^5 - 1\right)^5\,dx = \frac{\left(x^5 - 1\right)^6}{30}.
$$

**Step 4. Write the general solution and apply the initial condition.**

The general solution is:
$$
y(x) = \frac{\left(x^5 - 1\right)^6}{30} + C.
$$

Given $$y(1) = 5$$, substitute $$x = 1$$:
$$
y(1) = \frac{\left(1^5 - 1\right)^6}{30} + C = \frac{0^6}{30} + C = C.
$$
Thus:
$$
C = 5.
$$

**Final Answer.**

The solution to the initial value problem is:
$$
y(x) = \frac{\left(x^5 - 1\right)^6}{30} + 5.
$$
The solution to the initial value problem is: $$ y(x) = \frac{\left(x^5 - 1\right)^6}{30} + 5. $$

Solve the initial value problem.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions