First-order nonlinear differential equation

In summary, the student is trying to solve a first order nonlinear equation. They do not know how to solve the equation and need help.
  • #1
bennyh
4
0
Homework Statement
first order non linear equation
Relevant Equations
dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4

a,b,z,v,u are constant
t0=0 , T=T0
Homework Statement: first order non linear equation
Homework Equations: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4

a,b,z,v,u are constant
t0=0 , T=T0

Hi,
i need find an experession of T as function of t from this first order nonlinear equation:

dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4

a,b,z,v,u are constant
t0=0 , T=T0

i don't know how to solve this equation , tanks for helpers :)
 
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  • #2
bennyh said:
Homework Statement: first order non linear equation
Homework Equations: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4

a,b,z,v,u are constant
t0=0 , T=T0

Homework Statement: first order non linear equation
Homework Equations: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4

a,b,z,v,u are constant
t0=0 , T=T0

Hi,
i need find an experession of T as function of t from this first order nonlinear equation:

dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4

a,b,z,v,u are constant
t0=0 , T=T0

i don't know how to solve this equation , tanks for helpers :)
Welcome to the PF. :smile:

We require that you show some effort on your schoolwork problems before we can offer tutorial help.

What kind of DE solution methods have you learned so far, and do you see any that might be applicable here?

Also, it will help in the future if you learn to post using the LaTeX math editing features of the PF. There is a tutorial for how to post equations in LaTeX under INFO, Help at the top of the page. :smile:
 
  • #3
thank you for your response , i tried solve it with Bernoulli and Riccati Equations but some how it doesn't look normal to me due to the Riccati equation (that more general) have simple form of ##y'+py=fy^n##
bact to my equation :
##dT/dt-bT=a-Z[1/(1+vt)^2]-uT^4 ##
y=f(x) -> P(x)=b (constant num) and f(x)=-u (constant) and n=4
but i don't know how to treat a-Z[1/(1+vt)^2] in the formula.

need help :)
 
Last edited by a moderator:
  • #4
?
 
  • #5
I have the following suggestions. You have (rearranging terms and where primes on variables indicate differentiation w.r.t time),$$T' + bT=-uT^4 +f'(t) $$ where $$ f'(t) = a- \frac{Z}{(1+vt)^2}$$ Divide both sides of the equation by ##T^4## and rearrange to get,$$T^{-4}(T'-f'(t)) + bT^{-3}=-u$$ Now make the substitution$$v(t)= \frac{-T^{-3}}{3} - f(t)
\\ v'(t)=T^{-4} - f'(t)
\\T^{-3}= -3v(t)-3f(t)
$$We now have, after rearrangement, $$v'(t) -3bv(t)=-u + 3f(t)$$ which is in a form to apply an integration factor as outlined here,http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx. With,$$p(t)=-3b
\\g(t)=-u + 3f(t)$$ from Paul's above discussion on integration factors we have$$ v(t)=\frac{\int e^{-3bt}(-u +3f(t))dt + C}{e^{-3bt}}
\\T=\frac{1}{(-3v(t) -3f(t))^{\frac{1}{3}}}$$
 
Last edited:
  • #6
Please disregard my above post. I was totally wrong. The substitution I suggested is wrong. I feel like a fool and I apologize.
 
  • Like
Likes bennyh
  • #7
Fred Wright said:
Please disregard my above post. I was totally wrong. The substitution I suggested is wrong. I feel like a fool and I apologize.
thanks god that you wrong cause i don't understand it :) .
Do you have any other suggestion for this equation how to solve it?
 
Last edited by a moderator:

1. What is a first-order nonlinear differential equation?

A first-order nonlinear differential equation is an equation that involves an unknown function and its derivatives of only one variable. It is nonlinear because the unknown function and its derivatives are raised to powers or multiplied together.

2. How is a first-order nonlinear differential equation different from a first-order linear differential equation?

A first-order linear differential equation involves an unknown function and its derivatives of only one variable, but the unknown function and its derivatives are not raised to powers or multiplied together. This makes the equation linear, and it can be solved using standard techniques. Nonlinear equations, on the other hand, require more specialized methods for solving.

3. What is the order of a differential equation?

The order of a differential equation is the highest derivative of the unknown function that appears in the equation. For example, a first-order differential equation has only the first derivative, while a second-order equation has the second derivative as well.

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

Solving a first-order nonlinear differential equation involves finding a function that satisfies the equation. This can be done analytically, using techniques such as separation of variables, substitution, or integrating factors. It can also be solved numerically using computer software or approximation methods.

5. What are some real-life applications of first-order nonlinear differential equations?

First-order nonlinear differential equations have many practical applications in fields such as physics, engineering, economics, and biology. They can be used to model growth and decay processes, population dynamics, chemical reactions, and electrical circuits, among others. They also play a significant role in understanding complex systems and predicting their behavior.

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