Question

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.

$\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\left(x+y\right)$

Hint: Let $\mathit{u}\mathbf{=}\mathit{x}\mathbf{+}\mathit{y}$; then $\mathit{u}\mathbf{\text{'}}\mathbf{=}\mathbf{1}\mathbf{+}\mathit{y}\mathbf{\text{'}}$.

Short answer

Answer

The general solution of the differential equation is.

Step-by-step solution

Given information

The given differential equation is $y\text{'}=\cos \left(x+y\right)$

Definition of differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself to its derivatives of various orders.

Solve the differential equation

Assume $u=x+y$. Then, $u\text{'}=1+y\text{'}$. So the differential equation becomes,

$$\begin{gathered} u\text{'}-1=\cos u \\ u\text{'}=\cos u+1 \end{gathered}$$

, which has become a separable equation and its solution of is,

$$\begin{gathered} \frac{du}{\cos u+1}=dx \\ \tan \frac{u}{2}=x+C \\ u=2{\tan }^1\left(x+C\right) \end{gathered}$$

Substitute $u=x+y$

Therefore, $y=2{\tan }^{-1}\left(x+C\right)-x$