Nonlinear first order differential equation

In summary, the conversation was about solving the differential equation dx/dt + Rx^2 + G = 0 with given constants G and R and initial condition x(0) = 10^8. The approach of using Bernoulli and Laplace transformations was discussed, but ultimately it was determined that the equation is separable and can be solved by factoring and integrating. The integral was recognized as an arctangent.
  • #1
dreamwere
2
0

Homework Statement



Solve the diferential equation: dx/dt + Rx^2 + G = 0

G constant = 10^18
R constant = 10^-10

Initial conditions x(0) = 10^8

Homework Equations



what approach to take?

The Attempt at a Solution



First I try to apply bernoulli, but since in this equation I do not have a term x and since also the constat G is different to cero, it was not possible.

Secondly I try to solve in the frequency domain, and using laplace transformation I obtain:

Xs = 2R/s^2 + sx(0) + G

But then, things get complicate trying to convert to time domain for the second and first term becouse the first one implies a delta function in t = 0 and the second one implies a derivations of the inverse transfor of x(0), that is again an impulse sus the derivative of the impulse.

Thank you in advance for taking your time to help me.
 
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  • #2
That equation is separable.
 
  • #3
As vela said, the equation separable:

Write it as
[tex]\frac{dx}{Rx^2+ G}= dt[/tex]
and integrate both sides.

It might help to factor the "G" out leaving
[tex]\frac{1}{G}\frac{dx}{\frac{R}{G}x^2+ 1}= dt[/tex]
and recognize the integral as an "arctangent".
 
  • #4
Thank you done.

Sory for wasting your time, I should try harder before asking.
 

1. What is a nonlinear first order differential equation?

A nonlinear first order differential equation is a mathematical equation that involves the derivative of an unknown function with respect to one independent variable, and the function itself is raised to a power or multiplied by a function. This makes the equation nonlinear, meaning that the dependent variable and its derivatives are not directly proportional to each other.

2. What is the difference between a linear and nonlinear first order differential equation?

In a linear first order differential equation, the dependent variable and its derivatives are directly proportional to each other. This means that the equation can be written in the form y' + p(x)y = q(x), where p(x) and q(x) are functions of x. In contrast, a nonlinear first order differential equation does not follow this form and can be more challenging to solve.

3. What are some real-world applications of nonlinear first order differential equations?

Nonlinear first order differential equations are commonly used in physics and engineering to model real-world phenomena such as population growth, chemical reactions, and electrical circuits. They can also be used to predict the behavior of complex systems, such as weather patterns and biological processes.

4. How do you solve a nonlinear first order differential equation?

Solving a nonlinear first order differential equation often involves using analytical or numerical methods. Analytical methods involve manipulating the equation to find an exact solution, while numerical methods use algorithms and computer software to approximate a solution. The specific method used will depend on the complexity of the equation and the desired level of accuracy.

5. What are some challenges in solving nonlinear first order differential equations?

Nonlinear first order differential equations can be more difficult to solve than linear equations because they do not follow a standard form and may require more complex mathematical techniques. Additionally, some nonlinear equations have no closed-form solution and must be solved numerically. The initial conditions for the equation can also greatly affect the difficulty of finding a solution.

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