Question

$\mathit{y}\mathit{d}\mathit{y}\mathbf{+}\left(x{y}^2-8x\right)\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathit{y}\mathbf{=}\mathbf{3}$ when $\mathit{x}\mathbf{=}\mathbf{1}$.

Short answer

Answer:

The solution containing one arbitrary constant is $In\left(y^2-8\right)+x^2=C$, and with boundary condition, the value of the constant is C = 0. The particular solution is $In\left(y^2-8\right)+2^2=1$, and the plot of a slope field and some solution curves are as follows.

Step-by-step solution

Given Information

The given differential equation is$ydy+\left(x{y}^2-8x\right)dx=0$ with boundaries y = 3 when x = 1.

Definition of Differential Equation

A differential equation is a mathematical equation that connects one or more unknown functions with their derivatives. The study of differential equations primarily entails looking at their solutions (the set of functions that satisfy each equation) and characteristics.

Separate the variables

Separate the variables in $ydy+\left(x{y}^2-8x\right)dx=0$ and keep y on one side and x on the other.

$$\begin{gathered} ydy+\left(x{y}^2-8x\right)dx=0 \\ ydy=-x\left(y^2-8\right)dx \\ \frac{y}{y^2-8}dy=-xdx \end{gathered}$$

Integrate the differential equation

Integrate the above differential equation using the variable separable form.

Substitute,

Solve the separated form of differential equation with an arbitrary constant.

Re-substitutein the above equation.

…… (1)

Find the value of the arbitrary constant and particular solutions

Put the boundary conditions,when, in (1) and find the value of.

Substitute the value ofin (1) and find the particular solution.

Plot the slope field

Plot the slope field of and some solution curves.

Therefore, for the differential equation , the solution containing one arbitrary constant is , and with boundary condition, when . ,The value of constant is , and the particular solution is ., The plot of a slope field and some of solution curves are as follows.,