Solving ode with complex numbers

In summary, an ODE is a mathematical equation that involves an unknown function and its derivatives, while complex numbers are numbers with both a real and imaginary part. Complex numbers can be used to solve ODEs by converting the equation into a system of equations and have the advantage of allowing for a greater range of solutions and simplifying the solving process. However, there are limitations to using complex numbers, such as needing linear equations with constant coefficients and potentially more difficult to interpret solutions.
  • #1
cragar
2,552
3
I want to solve [itex] y''+y'+y=(sin(x))^2 [/itex] and try to use
[itex] y=Ae^{ix} [/itex] but then when I square it I get [itex] A^2 e^{2ix} [/itex]
I found y' and y'' and solved for A and it didn't work I guess I could use the formula for reducing powers but I would like to try and get around that.
 
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  • #2
hint
$$(D^3+4D)(\sin(x))^2=0$$
or
(sin(x))^2 is a solution of y'''+4y

chose the particular solution from those solutions
 

1. What is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that involves an unknown function and its derivatives. It describes the relationship between the function and its derivatives at different points.

2. What are complex numbers?

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part (with i being the imaginary unit).

3. How are complex numbers used to solve ODEs?

Complex numbers can be used to solve ODEs by converting the equation into a system of equations involving complex numbers. The solutions to these equations can then be used to determine the solution to the original ODE.

4. What are the advantages of using complex numbers to solve ODEs?

Using complex numbers to solve ODEs allows for a greater range of solutions, as they can give both real and complex solutions. It also simplifies the process of solving higher-order ODEs, as they can be converted into a system of first-order equations.

5. Are there any limitations to using complex numbers to solve ODEs?

Yes, there are some limitations to using complex numbers to solve ODEs. The equations must be linear and have constant coefficients in order for complex numbers to be used effectively. Additionally, the solutions may involve complex numbers, which can be more difficult to interpret than real numbers.

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