Question

Using (3.9) , 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(3.9) , and Example 1 .

${\mathbf{\text{y}}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{\text{ycosx}}\mathbf{=}\mathbf{\text{sin2x}}$

Short answer

The general solution of the differential equations is $y=2(\sin x-1)+C{e}^{-\sin x}$

Step-by-step solution

Given Information.

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

Step 2: Meaning of the first-order differential equation.

A first-order differential equation is defined by two variables, x and y , and its function f(x,y) is defined on an XY -plane region.

Find the general solution.

Write this differential equation to make it in the form$y^{\text{'}}+Py=Q$, from eq. 3.4

$$\begin{gathered} I=\int \cos xdx \\ =\sin x{e}^I \\ =e^{\sin x} \end{gathered}$$

Find a solution for this differential equation

$$\begin{gathered} y{e}^I=\int \left(\sin 2x\right)e^{\sin x}dx \\ =2e^{\sin x}(\sin x-1)+C \\ y=2(\sin x-1)+C{e}^{-\sin x} \end{gathered}$$

Therefore, the general solution of the differential equations is$y=2(\sin x-1)+C{e}^{-\sin x}$ .