Solving a differential equation

In summary: This is accomplished by setting up an equation in the x-variable, and solving for x.##\text{If you know the functions A, B, C, D, then you can set up the equation in the x-variable, and solve for x.}##
  • #1
zinDo
2
0
I want to solve Equation (1). w is a constant:
\begin{eqnarray} \text{Equation (1): }\frac{dx}{dt}=1+x^2w^2\end{eqnarray}
and I have been told that it is solved by (2):
\begin{eqnarray} \text{Equation (2): }x(t)=\frac{Ax(0)+B}{Cx(0)+D}\end{eqnarray}

Problem

I believe them, but before I keep solving it I want to know how one concludes that (1) is solved by (2). In order to get (2), should I integrate something like (3)? Is it some kind of ansatz that I should have known? ...
\begin{eqnarray} \text{Equation (3): }\frac{dx}{1+x^2w^2}=dt\end{eqnarray}
I don't know what is it that I should be looking for. Could I get a clue?
Thank you very much
 
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  • #2
zinDo said:
I want to solve Equation (1). w is a constant:
\begin{eqnarray} \text{Equation (1): }\frac{dx}{dt}=1+x^2w^2\end{eqnarray}
and I have been told that it is solved by (2):
\begin{eqnarray} \text{Equation (2): }x(t)=\frac{Ax(0)+B}{Cx(0)+D}\end{eqnarray}
This doesn't make sense to me, assuming that I'm interpreting your notation correctly. x(0) is just a constant, so what you have on the right side in equation 2 is just a constant (doesn't depend on t). It couldn't possibly be a solution to your diff. equation. x(t) = K, so dx/dt = 0, and 1 + x2w2 ≥ 1.
zinDo said:
Problem

I believe them, but before I keep solving it I want to know how one concludes that (1) is solved by (2). In order to get (2), should I integrate something like (3)? Is it some kind of ansatz that I should have known? ...
\begin{eqnarray} \text{Equation (3): }\frac{dx}{1+x^2w^2}=dt\end{eqnarray}
I don't know what is it that I should be looking for. Could I get a clue?
I would use a trig substitution to integrate what you have on the left side just above.
 
  • #3
##\int (\frac{1}{1+x^2}dx)=tan^{-1}(x) +C## , one of the basic integrals.

Eq. (2) is true if A, B, C, D are functions of t.
 

What is a differential equation?

A differential equation is an equation that relates an unknown function to its derivatives. It describes how a function changes over time or space and is commonly used in fields such as physics, engineering, and economics to model real-world phenomena.

Why is it important to solve a differential equation?

Solving a differential equation allows us to find the function that satisfies the equation and accurately describe the behavior of a system. This is crucial in understanding and predicting the behavior of various physical systems, from simple motion to complex systems like weather patterns.

What methods can be used to solve a differential equation?

There are several methods for solving a differential equation, including separation of variables, substitution, and using integration techniques such as Euler's method or Runge-Kutta methods. The most appropriate method depends on the type and complexity of the equation.

What are the applications of solving differential equations?

Solving differential equations has a wide range of applications in various fields, including physics, engineering, economics, biology, and chemistry. It can be used to model and predict the behavior of systems such as population growth, chemical reactions, and electrical circuits.

What are the challenges in solving a differential equation?

Solving a differential equation can be challenging due to the complexity of the equations and the wide range of methods that may need to be used. Additionally, some equations may not have an exact solution, and numerical approximations may be necessary. It also requires a strong understanding of calculus and mathematical concepts.

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