Find the general solution of the differential equation. (Use for the cons

Ask by Ramirez Mccarthy. in the United States

Mar 25,2025

Question

Find the general solution of the differential equation. (Use $$ C $$ for the cons $$ y=y^{\prime}=\sqrt{	an (x)} \sec ^{4}(x) $$

UpStudy AI · Accepted Answer

We start with the differential equation
$$
\frac{dy}{dx}=\sqrt{\tan(x)}\,\sec^4(x).
$$
Our goal is to find the general solution by integrating the right‐hand side.

**Step 1. Substitute $$ u=\tan(x) $$.**

We compute the derivative:
$$
\frac{du}{dx}=\sec^2(x) \quad \Longrightarrow \quad dx=\frac{du}{\sec^2(x)}.
$$
Substitute into the integral for $$ y $$:
$$
y=\int \sqrt{\tan(x)}\,\sec^4(x)\,dx = \int \sqrt{u}\,\sec^4(x)\,\frac{du}{\sec^2(x)}.
$$
This simplifies to
$$
y=\int \sqrt{u}\,\sec^2(x)\,du.
$$

**Step 2. Express $$\sec^2(x)$$ in terms of $$ u $$.**

Recall that
$$
\sec^2(x)=1+\tan^2(x)=1+u^2.
$$
So the integral becomes
$$
y=\int \sqrt{u}\,(1+u^2)\,du.
$$

**Step 3. Expand and integrate.**

Write the integrand in exponent form:
$$
\sqrt{u}\,(1+u^2)=u^{\frac{1}{2}}+u^{\frac{5}{2}}.
$$
Now, integrate term by term:
$$
\int u^{\frac{1}{2}}\,du=\frac{u^{\frac{3}{2}}}{\frac{3}{2}}=\frac{2}{3}\,u^{\frac{3}{2}},
$$
$$
\int u^{\frac{5}{2}}\,du=\frac{u^{\frac{7}{2}}}{\frac{7}{2}}=\frac{2}{7}\,u^{\frac{7}{2}}.
$$
Thus,
$$
y=\frac{2}{3}\,u^{\frac{3}{2}}+\frac{2}{7}\,u^{\frac{7}{2}}+C.
$$

**Step 4. Substitute back $$ u=\tan(x) $$.**

Replace $$ u $$ with $$ \tan(x) $$ to obtain:
$$
y=\frac{2}{3}\,\tan^{\frac{3}{2}}(x)+\frac{2}{7}\,\tan^{\frac{7}{2}}(x)+C.
$$

This is the general solution of the given differential equation.
The general solution of the differential equation $$ y = y' = \sqrt{\tan(x)} \sec^4(x) $$ is: $$ y = \frac{2}{3} \tan^{\frac{3}{2}}(x) + \frac{2}{7} \tan^{\frac{7}{2}}(x) + C $$ where $$ C $$ is the constant of integration.

Find the general solution of the differential equation. (Use for the cons

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