I know nothing about diff. equations, is this one solvable?

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In summary: Otherwise, it's better to stick to the method used to solve it in the first place.In summary, the conversation discusses a differential equation with an integral involving a squared function. The question is whether the equation can be solved by differentiating both sides and using Laplace transform. It is concluded that yes, this can be done, but since the equation is not linear, it is better to use the method of directly solving the integral equation. The importance of bringing back the result into the original equation is also mentioned. The non-linearity of the equation is due to the presence of a squared function, making it unsuitable for Laplace transform unless one is familiar with Laplace convolution.
  • #1
tamtam402
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I stumbled upon this diff. equation, which is a function of speed (v). Is there a technique I can use to solve it?

av(t) = b - c∫v(t)^2dt - d∫v(t)

Can I simply differentiate on both sides to get rid of the integral of the squared function, then use Laplace? The integrals go from 0 to an arbitrary time t.

Thanks!
 
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  • #2
tamtam402 said:
Can I simply differentiate on both sides to get rid of the integral of the squared function, then use Laplace?
Yes, you can. This will transform the integral equation to an ODE. But don't use Laplace transform because the ODE is not linear.
It's an ODE of the "separable" kind, directly leading to an integral easy to solve.
Don't forget to bring back your result into the original equation (not into the ODE) in order to compute the unknown constant.
 
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  • #3
It's not a differential equation, but an integral equation (yes, they have those, too._
 
  • #4
Thanks for the reply guys! I figured out how to solve it by deriving on both sides, putting the v's and dv on one side and the dt on the other side then integrating again.

I still have a question. What made the first equation non-linear? I'm not sure why Laplace cannot be used here.

Thanks!
 
  • #5
tamtam402 said:
What made the first equation non-linear? I'm not sure why Laplace cannot be used here.
In one of the terms, there is the square of the sought function. As a consequence the equation is not linear.
Try to use the Laplace transform of the square of a function. It involves Laplace convolution. If you know how to cope with it, then you can use Laplace transform.
 

1. Can differential equations be solved?

Yes, differential equations can be solved. In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. It is used to describe the behavior of systems in terms of their changing variables over time or space. There are various methods for solving differential equations, such as separation of variables, substitution, and using series solutions.

2. Are there different types of differential equations?

Yes, there are several types of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve one independent variable, while PDEs involve multiple independent variables. SDEs involve random or probabilistic elements in their equations.

3. Do I need to know advanced mathematics to solve differential equations?

Yes, solving differential equations requires a good understanding of advanced mathematics, including calculus, linear algebra, and differential equations themselves. However, there are also tools and software available that can help solve differential equations without needing to manually solve them.

4. Can differential equations be solved analytically or numerically?

Yes, differential equations can be solved analytically or numerically. Analytical solutions involve finding an exact mathematical expression for the solution, while numerical solutions involve approximating the solution using numerical methods and algorithms. The choice between analytical or numerical methods depends on the complexity of the equation and the desired level of accuracy.

5. Are differential equations used in real-world applications?

Yes, differential equations have numerous applications in various fields, including physics, engineering, economics, and biology. They are used to model and predict the behavior of systems and processes, including population growth, chemical reactions, and electrical circuits. Differential equations are also widely used in computer simulations and data analysis.

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