What is this problem asking for?

kalish

Member
I would like to know what exactly this problem is asking for. Also, if I'm on the right track.

**Problem:** A model for transport of a solute (moles of salt) and solvent (volume of water) across a permeable membrane has the form $$\dot{W}=A(k-\frac{M}{W}),\dot{M}=B(k-\frac{M}{W})$$

where $k$ is a parameter representing the bulk solute concentration and $A$ and $B$ are [parameters that represent the permeability of the membrane.

(a) The water volume $W$ is a positive quantity. **Show that the system can be made linear by a reparametrization.** ??

(b) Determine the transformation between solutions of the linear and nonlinear systems.

What does it mean by transformation?

Thanks.

I have crossposted this question on: differential equations - What is this problem asking for? - Mathematics Stack Exchange

Prove It

Well-known member
MHB Math Helper
I would like to know what exactly this problem is asking for. Also, if I'm on the right track.

**Problem:** A model for transport of a solute (moles of salt) and solvent (volume of water) across a permeable membrane has the form $$\dot{W}=A(k-\frac{M}{W}),\dot{M}=B(k-\frac{M}{W})$$

where $k$ is a parameter representing the bulk solute concentration and $A$ and $B$ are [parameters that represent the permeability of the membrane.

(a) The water volume $W$ is a positive quantity. **Show that the system can be made linear by a reparametrization.** ??

(b) Determine the transformation between solutions of the linear and nonlinear systems.

What does it mean by transformation?

Thanks.

I have crossposted this question on: differential equations - What is this problem asking for? - Mathematics Stack Exchange
I'd approach the problem like this, since both have a factor of \displaystyle \begin{align*} \left( k - \frac{M}{W} \right) \end{align*}, dividing gives

\displaystyle \begin{align*} \frac{\frac{dW}{dt}}{\frac{dM}{dt}} &= \frac{A \left( k - \frac{M}{W} \right) }{ B \left( k - \frac{M}{W} \right) } \\ \frac{dW}{dM} &= \frac{A}{B} \\ \frac{dW}{dM} &= C \textrm{ where }C = \frac{A}{B} \end{align*}

which is a(n almost trivially) linear ODE.