Practice Separation of Variables in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A method for solving first-order DEs of the form \frac{dy}{dx} = f(x) \cdot g(y): rearrange to \frac{dy}{g(y)} = f(x)\,dx, then integrate both sides.
If the rate of change factors into a piece that depends only on x and a piece that depends only on y, you can sort them onto opposite sides of the equation—all the y-stuff on the left, all the x-stuff on the right—then integrate each side in its own variable.
Example 1
easySolve dy/dx = xy with y(0)=2.
Example 2
hardSolve the logistic DE dy/dt = y(1-y) with y(0) = 1/2.
Example 3
easySolve dy/dx = x/y with y(0)=3.
Example 4
mediumSolve Newton's cooling: dT/dt = -0.1(T-20), T(0)=80.