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.

9 $\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{y}\mathbf{,}$ $\mathbf{y}\mathbf{=}\mathbf{1}$When$\mathbf{x}\mathbf{=}\mathbf{1}$

Short answer

The general solution is $y{e}^y=C_1{e}^x$and the particular solution is $y{e}^y=e^x$

Step-by-step solution

Given Information

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

Definition of Separable Differentiable 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 dxand terms involving xcan be put on one side and dy,and terms involvingon other. For example,$\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}\mathbf{=}\mathbf{g}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{dy}$.

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

$$\begin{gathered} \left(1+y\right)\frac{dy}{dx}=y \\ \frac{\left(1+y\right)}{y}dy=dx \end{gathered}$$

For the general solution we integrate both sides

$$\begin{gathered} \int \frac{1}{y}+1dy=\int dx \\ \ln \left(y\right)+y=x+c \end{gathered}$$

So, the general solution is

$y{e}^y=C_1{e}^x$

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$C_1$when the boundary condition$y=1$ when $x=1$ is satisfied is

$$\begin{gathered} e=C_{1}e \\ {C}_1=1 \end{gathered}$$

The particular solution is

$y{e}^y=e^x$

Draw the Slope field

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

$y\text{'}=\frac{y}{\left(1+y\right)}$