Nonseparable Differential Equation

In summary, the conversation discusses solving a differential equation that is not separable. Various methods are suggested, including substitution and finding an integrating factor. Eventually, it is discovered that there is a simple solution by setting u=y+x and using the method of separation, but a more general solution can be obtained by adding a constant to the particular solution.
  • #1
altcmdesc
66
0
I know that this differential equation is not separable, but is there a way to solve it?

dy/dx=y+x

I've tried a substitution of y=vx:

(dv/dx)x+v=x+vx
(dv/dx)x=x+vx-v
dv/dx=1+v-(v/x)

I'm stuck trying to rewrite that as a product of v and x.

Any help is appreciated!
 
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  • #2
Set u=y+x,

then you get:
du/dx=dy/dx+1,

whereby your diff.eq reads:

du/dx=1+u, which IS separable.

Alternatively, find an integrating factor to your diff.eq.
 
  • #3
By using that method, my answer was: eˣ-x-1.

Correct?

Thanks!
 
  • #4
Of course there are many methods to solve the above equations as they are a system of linear differential equations. The methods solved above are great for their simplicity but not so great in terms of generality.
 
  • #5
altcmdesc said:
By using that method, my answer was: eˣ-x-1.

Correct?

Thanks!
Why not check it out?

We have: y=e^x-x-1,

whereby:

dy/dx=e^x-1=(e^x-x-1)+x=y+x

so that is indeed A solution.

You still lack the general solution.
 
  • #6
isn't the general solution, the same thing but with the constant not being defined?

if not then please clarify it for me.
 
  • #7
AhmedEzz said:
isn't the general solution, the same thing but with the constant not being defined?

if not then please clarify it for me.
Yes it is...but your solution was [itex]y=e^x-x-1[/itex] which is a particular solution. the general sol'n would be [itex]y=e^x-x-1+C[/itex]...C= a constant. Never forget the constant of integration.
 
  • #8
The general solution is:

[tex]y(x)= A\cdot e^x -x- 1[/tex]

If you set A=1 then you get the particular solution of altcmdesc. However, the general solution is also obtained via the method of Arildno.

Nothing to do with adding a constant just like that, rock.freak667, you have to add it at the right place.
 

1. What is a nonseparable differential equation?

A nonseparable differential equation is a type of differential equation in which the dependent variable and the independent variable cannot be separated through algebraic manipulation. This means that the equation cannot be written in the form of y = f(x) where y is the dependent variable and x is the independent variable.

2. How do you solve a nonseparable differential equation?

Solving a nonseparable differential equation requires the use of advanced mathematical techniques such as substitution, integration, or series solutions. It may also involve using numerical methods such as Euler's method or Runge-Kutta method.

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

Nonseparable differential equations are commonly used in fields such as physics, engineering, and economics to model various physical phenomena, such as motion, heat transfer, population growth, and chemical reactions.

4. What are some common techniques used to simplify nonseparable differential equations?

Some common techniques used to simplify nonseparable differential equations include substitution, integration by parts, and partial fractions. These techniques help to reduce the complexity of the equation and make it easier to solve.

5. What is the difference between separable and nonseparable differential equations?

The main difference between separable and nonseparable differential equations is that separable equations can be written in the form of y = f(x), while nonseparable equations cannot. This means that separable equations can be solved by separating the variables and integrating each side, while nonseparable equations require more advanced techniques for solving.

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