Differential Equation problem, I think i solved it, just need it to be check

In summary, the conversation discusses a problem given by a teacher as extra credit. The problem involves finding a solution for the equation (x+y)dx-(x-y)dy=0 using substitution and polar coordinates. The conversation also mentions rearranging the equation and integrating both sides to obtain a final solution. The speaker is unsure if their approach is correct and asks for feedback.
  • #1
Pr0x1mo
21
0
My teacher gave me this as an extra credit question:

solve: (x+y)dx-(x-y)dy=0

I first used the substitution of y=vx or v=y/x

Taking the partial derivative of y = vx yields:

dy/dx = v + x(dv/dx)

So then i rearranged the equation like: dy/dx = (x+y)/(x-y) which equals:

v + xv' = (x+y)/(x-y)

Then i divided everything on the right side by x to obtain:

v + xv' = (1 + (y/x))/(1 - (y/x)), and since y/x = v then i get:

v + xv' = (1 + v)/(1 - v)

I then multiply both sides of the equation by (1 - v) to get:

v - xvv' = 1 + v

-xvv' = 1

-vv' = 1/x which is really:

-v (dv/dx) = 1/x, so to separate i mutliply both sides by dx:

-v dv = 1/x dx

then integrate both sides of the equation:

-1/2 v2 = ln|x| + c

and since v = y/x i get:

-1/2(y/x)2 = ln|x| + c

Is this correct, or am i completely off track?
 
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  • #2
I haven't followed your text,
but I'd try going in polar coordinates. Looks promising.

[tex]
(x+y)dx+(x-y)dy =0\\

[/tex]

[tex]

(rcos\theta+rsin\theta)(drcos\theta-rsin\theta d\theta)+(rcos\theta-rsin\theta)(drsin\theta+rcos\theta d\theta)=0 \\
[/tex]

[tex]((cos\theta)^2-(sin\theta)^2+2sin\theta cos\theta)rdr + r^2((cos\theta)^2-(sin\theta)^2-2sin\theta cos\theta)d\theta =0\\
[/tex]

[tex]

((cos\theta)^2-(sin\theta)^2+2sin\theta cos\theta)rdr + r^2((cos\theta)^2-(sin\theta)^2-2sin\theta cos\theta)d\theta=0

[/tex]
 
  • #3
[tex]

\frac{dr}{d\theta} = r\frac{1-sin4\theta}{cos4\theta}\\

[/tex]

I get this but I don't know what to do next.
 

Related to Differential Equation problem, I think i solved it, just need it to be check

1. What is a differential equation?

A differential equation is a mathematical equation that describes how a variable changes over time. It involves derivatives, which represent the rate of change of the variable.

2. How do I solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some common techniques include separation of variables, integrating factors, and using power series. It is important to have a good understanding of calculus and algebra to solve differential equations.

3. Can I check my solution to a differential equation problem?

Yes, you can check your solution by substituting it back into the original differential equation and verifying that it satisfies the equation. You can also use numerical methods to approximate the solution and compare it to your solution.

4. What is the importance of differential equations in science?

Differential equations are used to model and understand the behavior of many natural phenomena in fields such as physics, engineering, and biology. They allow us to make predictions and solve real-world problems.

5. What are some common applications of differential equations?

Differential equations are used in various fields, such as calculating population growth, predicting the spread of diseases, modeling chemical reactions, and analyzing electrical circuits. They are also used in the development of computer graphics and animation.

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