# Dog chases his tail ! - "parametric differential/Integral equation"..?

I have the following situation where I am interested in the function $m(t)$

$$\frac{dm}{dt}=4T(t)^{3}+T(t)^{2}$$
$$T(\tau)=T_{0}-\int_{0}^{\tau}(\frac{dm}{dt})dt*Q_{S}$$

Is there a way to solve it analytically using Mathematica?
Andi

ps.: I guess I just don't know what to look for

• Take a look a DSolve - and maybe get familiar with Mathematica a bit.
– gwr
Commented May 10, 2016 at 9:46
• The notation is certainly confusing me. If I interpret $\frac{dm}{dt}(T)$ as the derivative of $m(T)$ with respect to $t$, then that's $0$; if that's the product of $\frac{dm}{dt}$ and $T$, then the right hand side is effectively a constant. So, what then? Commented May 10, 2016 at 9:53
• I hope it is more clear after that edit ... - m is the mass, T the temperature - mass changes depending on the Temperature. The temperature drops by losing mass (Sublimation) Commented May 10, 2016 at 9:58
• What do you get if you differentiate your second equation with respect to $t$? Commented May 10, 2016 at 10:02
• something arbitrary- but I just realized, that the Integral is written wrong - next edit ... Commented May 10, 2016 at 10:06

Following up on @J.M.'s observations in the comments, differentiate

$$m(t)=m(0)-\frac{T(t)-T_0}{Q_S}$$

to get $$\frac{dm}{dt}=-\frac{T'(t)}{Q_S}$$

Combine with

$$\frac{dm}{dt}=4 \, T(t)^{3}+T(t)^{2}$$

to get a differential equation in T[t]:

$$T'(t)=-\text{Qs} \left(4 \, T(t)^3+T(t)^2\right)$$

Use DSolve with initial value T[0] == t0:

DSolve[{T'[t] == -Qs ( 4 T[t]^3 + T[t]^2), T[0] == t0}, T, t]


• Thank you very much @kglr, obviously it was a mathematics/approach problem :-) - and my bachelor studies were too long ago, ... maybe then I could have handled it better Commented May 10, 2016 at 11:32
• @G.Andi, my pleasure. You were almost there:) Thank you for the accept.
– kglr
Commented May 10, 2016 at 11:35
• now i will do this approach with my real dm/dt function which is a hell compared to $\frac{dm}{dt}=4 \, T(t)^{3}+T(t)^{2}$ :-) Commented May 10, 2016 at 11:36