Question

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d u}=4(\sin u)^{3}(\cos u)$$

Short answer

The general solution for the given differential equation is \( y = \sin^{4}(u) + C \).

Step-by-step solution

Recognize the function

Recognize that \(\frac{d y}{d u}=4(\sin u)^{3}(\cos u)\) is the derivative of a function with respect to \(u.\) In particular, this is a composite function that is a product of 4, \(\sin(u)\) to the power of 3, and \(\cos(u)\).

Identify the technique to solve the problem

Identify the appropriate technique to solve this problem. In this case, the Integral of a function can be used to find the original function when its derivative is given. Hence, we need to integrate both sides of the given equation to find the function \(y\).

Application of integral

Apply the integral of both sides of the equation: \[\int \frac{d y}{d u} du = \int 4(\sin u)^{3}(\cos u) du\] Simplifying this will give: \[y = 4 \int (\sin u)^{3}(\cos u) du\]

Use substitution method

Since the integrand is a composite function, the substitution method can be used here. Setting \(v = \sin(u)\), the equation becomes: \[d v = \cos(u) du\] Then, the integral becomes: \[y = 4 \int v^{3} dv\]

Solve the integral

Evaluate the integral using the basic formula for the integral of \(v^n\), which is \(\frac{v^{n+1}}{n+1}\). In this case, \(n=3\), so: \[y = 4 \frac{v^{3+1}}{3+1} + C\] where \(C\) is the constant of integration. Hence, \[y = v^{4} + C\]

Substitute back the actual variable

Replace \(v\) with \(\sin(u)\) to express \(y\) as a function of \(u\): \[y = (\sin(u))^{4} + C\]

Simplify the function

Finally, simplify the function to get the general solution to the differential equation: \[y = \sin^{4}(u) + C\]