Question

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves.

$\mathbf{12}\mathbf{.}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{xy}\mathbf{)}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}$ y = 1When x = 1.

Short answer

The general solution is ${\mathrm{ye}}^{\mathrm{y}}=\frac{{\mathrm{C}}_1}{\mathrm{x}}$and the particular solution is${\mathrm{ye}}^{\mathrm{y}}=\frac{\mathrm{e}}{\mathrm{x}}$

Step-by-step solution

Given Information

We have given a differential equation $(\mathrm{x}+\mathrm{xy})\mathrm{y}\text{'}+\mathrm{y}=0$with the boundary condition y = 1 when x = 1.

Definition of Separable Differential equation

Any equation of the form$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{g}\mathbf{\left(}\mathbf{y}\mathbf{\right)}$is called separable that is any equation in which dx and terms involving xcan be put on one side and dy, and terms involvingon other. For example,

f (x) dx = g (y) dy.

Find the General Solution

Any solution of the differential equation containing linearly independent arbitrary constant is the general solution of the differential equation.

Let’s first start by separating the variables

$$\begin{gathered} \mathrm{x}(1+\mathrm{y})\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y} \\ -\frac{1+\mathrm{y}}{\mathrm{y}}\mathrm{dy}=\frac{\mathrm{dx}}{\mathrm{x}} \end{gathered}$$

For the general solution we integrate both sides

$$\begin{gathered} -\int \frac{1+\mathrm{y}}{\mathrm{y}}\mathrm{dy}=\int \frac{\mathrm{dx}}{\mathrm{x}} \\ -\left[\mathrm{In}\left(\mathrm{y}\right)+\mathrm{y}\right]=\ln \left(\mathrm{x}\right)+\mathrm{C} \end{gathered}$$

So, the general solution is

${\mathrm{ye}}^{\mathrm{y}}=\frac{{\mathrm{C}}_1}{\mathrm{x}}$

Find the Particular Solution

The solution obtained from the general solution by giving some particular values to the arbitrary constants is called particular solution.

The value of the constant ${\mathrm{C}}_1$when the boundary condition y = 1 when x = 1 is satisfied is

$$\begin{gathered} 1{\mathrm{e}}^1=\frac{{\mathrm{C}}_1}{1} \\ \mathrm{e}={\mathrm{C}}_1 \end{gathered}$$

The particular solution is

${\mathrm{ye}}^{\mathrm{y}}=\frac{\mathrm{e}}{\mathrm{x}}$

Draw the Slope field

Draw the slope field for this we will use the slope of the equation, which is

$\mathrm{y}\text{'}=\frac{\mathrm{y}}{(\mathrm{x}+\mathrm{xy})}$