First-order nonlinear differential equation

[bennyh]

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 :)

 

[berkeman]

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.

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.

 

[bennyh]

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 :)

 

[bennyh]

?

 

[Fred Wright]

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$$ v(t)=\frac{\int e^{-3bt}(-u +3f(t))dt + C}{e^{-3bt}}
\\T=\frac{1}{(-3v(t) -3f(t))^{\frac{1}{3}}}$$

 

[Fred Wright]

Please disregard my above post. I was totally wrong. The substitution I suggested is wrong. I feel like a fool and I apologize.

 

[bennyh]

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?