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. $ydy+(x{y}^2-8x)dx=0,y=3$ when $x=1$

Short answer

The particular solution is y = 3 x^{-8}. Apply boundary condition: C = 3.

Step-by-step solution

Rearrange Equation

Separate the variables by moving all terms involving y on one side and all terms involving x on the other side. Start with the given equation:y dy + (x y^2 - 8x) dx = 0Rewrite it as:y dy = - (x y^2 - 8x) dxOr equivalently as:y dy = -x y^2 dx + 8x dx.

Simplify and Separate Variables

Separate the equation by dividing both sides by y^2 and by x, ensuring each side of the equation only contains one variable:dy/y = (- y/x + 8/x) dxTherefore, we get:dy/y + y dx = -8 dx.

Integrate Both Sides

Integrate both sides of the equation to find the general solution:∫ dy/y = ∫ (-8/x) dxThis gives:ln|y| = -8 ln|x| + COr, in exponential form:y = C x^{-8}.

Apply Boundary Condition

Use the given boundary condition y(1) = 3 to find the constant C:3 = C (1)^{-8}Thus, C = 3.

Formulate Particular Solution

Substitute the value of C back into the general solution to obtain the particular solution:y = 3 x^{-8}.

Plot Slope Field and Solution Curves

Use a computer software or graphing tool to generate a slope field for the differential equation and plot the solution curve y = 3 x^{-8}.

Key concepts

Separation of Variables

Separation of variables is a fundamental technique used to solve differential equations. It involves rearranging the equation so that all the terms involving one variable are on one side, and all the terms involving the other variable are on the opposite side.

For the given equation, we started with:

$ydy+(x{y}^2-8x)dx=0$
We separated the variables by moving all terms involving $y$ to one side and all terms involving $x$ to the other side:

$ydy=-(x{y}^2-8x)dx$

This separation allowed us to integrate each side independently. When solving differential equations, it's crucial to ensure the variables are properly separated before proceeding to the next step, which is integration.

Integration

Integration is the next step after separating the variables. It involves finding the antiderivative of each side of the equation. This transforms the derivatives into functions.

From the separated equation:
$\frac{dy}{y}=-\frac{8}{x}dx$
we integrated both sides to find the general solution:
$\text{bigint}\frac{dy}{y}=\text{bigint}-\frac{8}{x}dx$
This gave us the integrated form:
$\text{ln}|y|=-8\text{ln}|x|+C$
Finally, we exponentiated both sides to solve for $y$:
$y=C{x}^{-8}$.

Integration turns the differential equation into an explicit algebraic form, which makes it easier to work with and apply boundary conditions.

Boundary Conditions

Boundary conditions are specific values given in a problem that allow us to find the particular solution of a differential equation. These conditions provide a specific point, often giving the value of the function at a particular value of $x$.

For our problem, we were given the boundary condition $y(1)=3$, meaning that when $x=1$, $y$ should be $3$.
We use this information to find $C$, the constant of integration. Substituting into the general solution we obtained:

$3=C(1{)}^{-8}$,
we solved for $C$ and found:

$C=3$.

Thus, our particular solution became:
$y=3x^{-8}$.

Boundary conditions are essential because they ensure the solution meets the specific criteria given in the problem.

Slope Field

A slope field, also known as a direction field, is a graphical representation that allows us to visualize the solutions of a differential equation. Each tiny line segment in a slope field represents the slope of the solution curve at that point.

For the differential equation:
$ydy+(x{y}^2-8x)dx=0$,
we can use computer software or a graphing tool to generate a slope field. This involves plotting short lines at various points $x$ and $y$, with each line's slope determined by the differential equation.

Slope fields are invaluable as they provide a visual guess for the behavior of solution curves without solving the equation analytically. They help us understand how different initial conditions can affect the shape and direction of solutions.

Solution Curves

Solution curves are the graphical representations of the solutions to a differential equation. Each curve corresponds to a particular set of initial conditions.

For our particular solution:
$y=3x^{-8}$
this function can be plotted on a slope field to visualize its behavior.

Solution curves help in understanding the trajectory of solutions over different ranges of the independent variable $x$. They give a clear representation of how the function $y$ changes as $x$ changes.

In practical terms, plotting several solution curves, each starting from different initial points, can show how the system behaves under various conditions, providing deeper insight into the nature of the differential equation.