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

$$m \cdot C_P \cdot {{dT_M } \over {dt}} = U \cdot A\left( {T_R - T_M } \right)$$

and

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

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.