Which type differential equation is this?

In summary, Nonlinear differential equation, First order, nonlinear, Integration factor may be necessary to solve, DSolve in Mathematica can't solve it, numerically or by fitting a curve, radius of convergence is difficult to compute.
  • #1
Susanne217
317
0
Which type differential equation is this??

I simply can't recognize it


[tex]y' = \frac{1}{3}y^{\frac{1}{2}} + t^{\frac{1}{3}}[/tex]

Which type of differential equation is??

non-linear?
 
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  • #3


Mark44 said:
First order, nonlinear.

Hi and thank you for your answer...

which method do I use to solve this? integration factor? or separation of the variables?

I keep ending wrong result :(

So if you could point me direction of the right method to solve it. That would be nice :)
 
  • #4


Susanne217 said:
So if you could point me direction of the right method to solve it. That would be nice :)

It's non-linear and often these require special techniques to solve. Integration factor is usually used for linear equations and this one can't be separated. This is what I usually do: try a bit to solve it by hand. I did that and got nothing. My next approach is to use DSolve in Mathematica:

DSolve[y'[t] == 1/3 y[t]^(1/2) + t^(1/3), y, t]

At least that way, if Mathematica gives me a solution, I know it's relatively easy to solve and the exact form of the answer often gives me a hint on how to solve it. However in this case Mathematica can't solve it. At that point, I think it's probably not easy to solve symbolically although sometimes Mathematica is in error. I may or may not look in a DE handbook. Sometimes that's helpful. Finally, my next approach would be to use NDSolve in Mathematica and solving it (an IVP) numerically and if necessary, fit a curve to the data if some approx. symbolic representation is sufficient.
 
  • #5


jackmell said:
It's non-linear and often these require special techniques to solve. Integration factor is usually used for linear equations and this one can't be separated. This is what I usually do: try a bit to solve it by hand. I did that and got nothing. My next approach is to use DSolve in Mathematica:

DSolve[y'[t] == 1/3 y[t]^(1/2) + t^(1/3), y, t]

At least that way, if Mathematica gives me a solution, I know it's relatively easy to solve and the exact form of the answer often gives me a hint on how to solve it. However in this case Mathematica can't solve it. At that point, I think it's probably not easy to solve symbolically although sometimes Mathematica is in error. I may or may not look in a DE handbook. Sometimes that's helpful. Finally, my next approach would be to use NDSolve in Mathematica and solving it (an IVP) numerically and if necessary, fit a curve to the data if some approx. symbolic representation is sufficient.

I get a solution in Maple, but it very strange containing integral sign etc. There must be away to solve this equation without having to use computeral power. Hallsofty we need your guidence Master Science Jedi...
 
  • #6


Susanne217 said:
Hallsofty we need your guidence Master Science Jedi...

Agreed, but I would say just "Master Jedi".
 
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  • #7


Hi Guys !

I am not a Jedi, so I let the most honorific job for them.
I did only a subaltern job which consists in expressing the result in terms of series development.
To be honest, I confess that my devoted computer did the even more subaltern work, which in fact is almost the whole, while I had a drink.
Well, have a look at the joint document.
 

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  • #8


Wait, let me check . . . yep, that's you alright. Startin' to look like it to me. Anyway, that's really nice Jacquelin. I think it would be nice to verify that solution, say numerically to some acceptable level of precision. And how does one compute the radius of convergence? I have problems figuring that out when a Cauchy product is involved.
 
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Question 1: What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is often used to model various physical phenomena in fields such as physics, engineering, and economics.

Question 2: What are the different types of differential equations?

There are several types of differential equations, including ordinary differential equations, partial differential equations, and exact differential equations. Ordinary differential equations involve only one independent variable, while partial differential equations involve multiple variables. Exact differential equations are a special type of ordinary differential equation that can be solved without an integrating factor.

Question 3: How do I know which type of differential equation I am dealing with?

The type of differential equation can be determined by looking at the order, linearity, and type of independent and dependent variables involved. Ordinary differential equations can be further classified as first order, second order, or higher order. Partial differential equations can be classified as elliptic, parabolic, or hyperbolic. Exact differential equations can be identified by the presence of an exact differential form.

Question 4: What is the purpose of solving differential equations?

Solving differential equations allows us to find the function that satisfies the given relationship and can be used to predict the behavior of a system. It is a powerful tool for understanding and modeling real-world phenomena in various fields of science and engineering.

Question 5: Are there any common techniques for solving differential equations?

Yes, there are several common techniques for solving differential equations, including separation of variables, integrating factors, variation of parameters, and using power series. The appropriate technique to use depends on the type and complexity of the differential equation.

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