Solving 2 Equations with ODE and Diff

[TSN79]

I'm trying to solve a set of two equations, one of which is an ODE. They are

<br /> m \cdot C_P \cdot {{dT_M } \over {dt}} = U \cdot A\left( {T_R - T_M } \right)<br />

and

<br /> Q_P + \rho \cdot C_P \cdot \dot V\left( {T_O - T_R } \right) = U \cdot A\left( {T_R - T_O } \right)<br />

I want to solve this set for T_M and T_R, but I'm not sure about the procedure, because of the diff. Any help will be appreciated

 

[HallsofIvy]

Are we to assume that T_m, T_R, and T₀ are functions of t? If T₀ is also unknown, then you don't have enough equations. If T₀ is a known function of t, then from
Q_P + \rho \cdot C_P \cdot \dot V\left( {T_O - T_R } \right) = U \cdot A\left( {T_R - T_O } \right)
Q_P - \rho \cdot C_P \cdot \dot V\left( {T_R - T_O } \right) = U \cdot A\left( {T_R - T_O } \right)
\left(U\cdot A+ \rho \cdot C_P \cdot \dot V\right)\left(T_R- T_O\right)= Q_P
T_R- T_O=\frac{Q_P}{U\cdot A+ \rho \cdot C_P \cdot \dot V\right}
T_R= T_O+ \frac{Q_P}{U\cdot A+ \rho \cdot C_P \cdot \dot V\right}
Now put that function into
m \cdot C_P \cdot {{dT_M } \over {dt}} = U \cdot A\left( {T_R - T_M } \right)
and solve the differential equation.