Question

In Exercises \(1-10,\) find the general solution to the exact differential equation. $$\frac{d y}{d t}=3 t^{2} \cos \left(t^{3}\right)$$

Short answer

The general solution to the exact differential equation \(\frac{d y}{d t}=3 t^{2} \cos \left(t^{3}\right)\) is \(y(t)= \sin \left(t^{3}\right) + c\), where \(c\) is an arbitrary constant.

Step-by-step solution

Identify the Type of Differential Equation

In this case, the differential equation is an ordinary differential equation that can be classified more specifically as a first order linear differential equation, because it only involves a function \(y\) and its first derivative. The differential equation is \(\frac{d y}{d t}=3 t^{2} \cos \left(t^{3}\right)\).

Solve the Differential Equation by Integration

This differential equation can be written in a more familiar way, as \(y'=3 t^{2} \cos \left(t^{3}\right)\). Here, \(y'\) is the derivative of \(y\), and the right side is a function of \(t\). The next step is to integrate both sides of the equation with respect to \(t\). The left side evaluates to \(y(t)\), while the right side will need to be computed using integral calculus. This can be visualized as: \(\int dy = \int 3 t^{2} \cos \left(t^{3}\right) dt\). Then, use the chain rule to simplify: let \(u=t^{3}\), so that \(du=3t^{2}dt\). It now simplifies to \(\int dy = \int \cos u du\). Therefore, \(y(t)= \sin u + c\), where \(c\) is the constant of integration. Replace \(u\) with \(t^{3}\) to get the solution: \(y(t)= \sin \left(t^{3}\right) + c\).

Present the General Solution

Recall that the 'general' solution to a differential equation is the solution that includes all possible solutions, which means it includes an arbitrary constant 'c'. In other words, the general solution to the original equation is: \(y(t)= \sin \left(t^{3}\right) + c\), for any value of \(c\).