Solve the differential equation xdy=(5y+x+1)dx

In summary, the conversation involved solving two differential equations and finding a differential equation with a given solution. The first equation required using substitution and rearranging terms, while the second equation required getting all x and dx terms on one side and all y and dy terms on the other. In the third part, the speaker was unsure of how to proceed after getting rid of the constant c_2.
  • #1
hbomb
58
0
I need someone to look over my work if possible

1) Solve the differential equation
xdy=(5y+x+1)dx

Here is what I did:
x=(5y+x+1)dx/dy
[tex]x=5xy+\frac{x^2}{2}+x[/tex]
[tex]0=5xy+\frac{x^2}{2}[/tex]
[tex]\frac{-x^2}{2}=5xy[/tex]
[tex]y=\frac{-x}{10}[/tex]

2) Solve: [tex]y(x^2-1)dx+x(x^2+1)dy=0[/tex]

Here is what I did:

[tex]x^2ydx-ydx+x^3dy+xdy=0[/tex]

[tex]\frac{xdy-ydx}{x^2}+ydx+xdy=0[/tex]

[tex]d(\frac{y}{x})+d(xy)=0[/tex]
This is where I am stuck at. I need to have something multiply the first differential that will yield something in the form of [tex]\frac{y}{x}[/tex] and also have the same thing multiply the second differential and will yield something in the form of xy.

3) Find a differential equation with the solution [tex]y=c_1sin(2x+c_2)[/tex]

Here is what I did:

[tex]y'=2c_1cos(2x+c_2)[/tex]

and since
[tex]c_1=\frac{y}{sin(2x+c_2)}[/tex]

[tex]y'=\frac{2ycos(2x+c_2)}{sin(2x+c_2}[/tex]

[tex]y'=2ycot(2x+c_2)[/tex]

I'm not sure what to do after this. I know I need to somehow get rid of the [tex]c_2[/tex] constant, but I don't know how to do this.
 
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  • #2
In the second one just get all of the x and dx terms on one side and all of the y and dy terms on the other. That should only take one step.

[tex]
y\left( {x^2 - 1} \right)dx + x\left( {x^2 + 1} \right)dy = 0
[/tex]

[tex]
\Rightarrow \left( {\frac{{x^2 - 1}}{{x^2 + 1}}} \right)dx = - \frac{1}{y}dy
[/tex]

Now integrate both sides.
 

1. What is a differential equation?

A differential equation is a mathematical equation that involves an unknown function and its derivatives. It describes the relationship between a function and its rate of change.

2. How do I solve a differential equation?

To solve a differential equation, you need to first identify the type of equation (e.g. linear, separable, exact) and then use appropriate methods and techniques to solve it. This may involve finding an integrating factor, using substitution, or separating variables.

3. What is the solution to the given differential equation?

The solution to a differential equation is the function that satisfies the equation for all values of the independent variable. In the case of xdy=(5y+x+1)dx, the solution is y = -x/5 - 1/25 + C, where C is a constant of integration.

4. Can I check my solution to the differential equation?

Yes, you can check your solution by substituting it into the original differential equation and verifying that it satisfies the equation. In addition, you can also use numerical methods or graphing to visualize the solution and see if it matches the behavior of the equation.

5. What are some real-world applications of differential equations?

Differential equations are used to model and solve problems in various fields such as physics, engineering, biology, economics, and more. Some examples include population growth, radioactive decay, motion of objects, and heat transfer.

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