Question

Using $\left(3.9\right)$, find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $\left(3.9\right)$, and Example 1.

$\mathit{y}\mathbf{\text{'}}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathbf{+}\mathit{y}\mathbf{=}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{x}$

Short answer

Answer:

The general solution of the differential equation is $y=\frac{x-\cos \left(x\right)+C}{sex\left(x\right)+\tan \left(x\right)}$.

Step-by-step solution

Given information

The given differential equation is$y\text{'}\cos x+y={\cos }^{2}x$.

Meaning of the first-order differential equation

A first-order differential equation is defined by two variables, x and y, and its function $\mathit{f}\left(x,y\right)$is defined on an XY-plane region.

Find the general solution

Write this differential equation to make it in the form $y\text{'}+Py=Q$; that is

$y\text{'}+ysec\left(x\right)=\cos \left(x\right)$

From equation 3.4,

$$\begin{gathered} I=\int sec\left(x\right)dx \\ =In\left[sec\left(x\right)+\tan \left(x\right)\right]e^I \\ =e^{In\left[sec\left(x\right)+\tan \left(x\right)\right]} \\ =sec\left(x\right)+\tan \left(x\right) \end{gathered}$$.

Find a solution for this differential equation.

Therefore, the general solution of the differential equation is.