# Combining two differential equations

#### alexander9408

Two quantities, x and y, vary with time t. State the differential equation for each of the following.
(a) The rate of change of y with respect to t varies inversely as x.
(b) The rate of change of x with respect to t varies inversely as y. Combining the differential equation of (a) and (b), form a differential equation involving only x and y, solve the differential equation expressing y in terms of x.

How do I combine the two differential equations into one?

#### Prove It

MHF Helper
Two quantities, x and y, vary with time t. State the differential equation for each of the following.
(a) The rate of change of y with respect to t varies inversely as x.
(b) The rate of change of x with respect to t varies inversely as y. Combining the differential equation of (a) and (b), form a differential equation involving only x and y, solve the differential equation expressing y in terms of x.

How do I combine the two differential equations into one?
Do you understand what's meant by inverse proportion? If "a" is inversely proportional to "b", then $$\displaystyle \displaystyle a = \frac{k}{b}$$, where "k" is a constant.

#### alexander9408

Yes, my problem was combining the two differential equations dy/dt = k/x and dx/dt = h/y (k and h are constants) into one differential equation and expressing y in terms of x.

#### Prove It

MHF Helper
Remember $$\displaystyle \displaystyle \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dx}$$, so

\displaystyle \displaystyle \begin{align*} \frac{\frac{dy}{dt}}{\frac{dx}{dt}} &= \frac{\frac{k}{x}}{\frac{h}{y}} \\ \frac{dy}{dx} &= \frac{k\,y}{h\,x} \end{align*}

This is a separable first order DE.