Solving a Differential Equation Conflict

  • Thread starter gulsen
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In summary, you had a typo in your equation where you forgot the integration constant and as a result got a wrong solution. You then solved the equation as a non-homogenous equation and got the correct solution.
  • #1
gulsen
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I have a basic differential equation:
[tex]\frac{dy}{dx} = x + y, y(0) = 1[/tex]

Now, when I try to solve this by making it exact
[tex]\mu \frac{dy}{dx} + \mu y = \mu x[/tex]
I get [tex]\mu = e^{-x}[/tex] and solution [tex]-x-1[/tex]. This doesn't satisfy the initial condition. But when I try to solve it as a non-homogenous equation as:
[tex]\frac{dy}{dx} + y= x[/tex]
I get
[tex]y_p = 2e^x, y_c = -x-1[/tex]
so
[tex]y = 2e^x-x-1[/tex]

Which seems to be a correct & full solution. What was I missing in the first try?
 
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  • #2
You are solving dy/dx + y = x which is wrong since the original equation is
dy/dx = x + y, or dy/dx - y = x so now u(x) = e^(-x) and so on and it works out.


And yes, the answer is y = 2e^x - x - 1
 
  • #3
gulsen said:
I have a basic differential equation:
[tex]\frac{dy}{dx} = x + y, y(0) = 1[/tex]
Now, when I try to solve this by making it exact
[tex]\mu \frac{dy}{dx} + \mu y = \mu x[/tex]
I assume this was just a typo, but that equation should be
[tex]\mu \frac{dy}{dx} - \mu y = \mu x[/tex]
I get [tex]\mu = e^{-x}[/tex] and solution [tex]-x-1[/tex]. This doesn't satisfy the initial condition.
Your error: you forgot the constant of integration. You should have obtained
[tex]y = -(x+1) + c/\mu = ce^x - (1+x)[/tex]
and then solved for the initial condition [itex]y(0)=1[/itex] yielding [itex]c=2[/itex] or
[tex]y = 2e^x -(x+1)[/tex]
But when I try to solve it as a non-homogenous equation as:
[tex]\frac{dy}{dx} + y= x[/tex]
I get
[tex]y_p = 2e^x, y_c = -x-1[/tex]
so
[tex]y = 2e^x-x-1[/tex]
You're nomenclature is backward here. The solution to the homogeneous equation is called the complementary function and is denoted as [itex]y_c[/itex]. The complementary function generally involves arbitrary constants. A solution to the inhomogeneous equation is called a particular function and is denoted as [itex]y_p[/itex].
In this case, the solution to the homogeneous equation [itex]y^\prime-y=0[/itex] is
[tex]y_c = ce^x[/tex]
where [itex]c[/itex] is an arbitrary constant and
[tex]y_p = -(x+1)[/tex]
is a particular solution to the inhomogeneous equation. Combining these,
[tex]y = ce^x - (1+x)[/tex]
which meets the initial conditions when [itex]c=2[/itex].
 
Last edited:
  • #4
Oh, that was just a typo. Thanks D_H, it ends up that I've forgotten the integration constant.
About naming, take it easy, I'm just a freshman!
 

1. What is a differential equation conflict?

A differential equation conflict is a problem that arises when attempting to solve a differential equation. It occurs when there is no unique solution or when the solution is not physically meaningful.

2. What causes a differential equation conflict?

There are several possible causes for a differential equation conflict, including errors in the initial conditions or boundary conditions, mistakes in the mathematical formulation, or limitations in the techniques used to solve the equation.

3. How can a differential equation conflict be resolved?

To resolve a differential equation conflict, it is important to carefully check and verify the initial and boundary conditions, as well as the mathematical formulation. If there are no errors, alternative techniques such as numerical methods or approximations may be used.

4. Are there any common strategies for solving differential equation conflicts?

Yes, there are several common strategies for solving differential equation conflicts. These include changing the form of the equation, using different mathematical techniques, or adjusting the initial and boundary conditions to eliminate the conflict.

5. Can differential equation conflicts be avoided?

In some cases, differential equation conflicts can be avoided by carefully choosing the mathematical formulation and initial/boundary conditions. However, in many cases, conflicts may be inevitable and must be resolved through careful analysis and problem-solving strategies.

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