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Chapter exploration

Differential equations

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Consider dV/dt=6−0.2VdV/dt=6-0.2V, where VV is in litres and tt in minutes. Compare initial volumes 0, 20 and 40 L. Predict each initial rate, whether the volume rises or falls, and any common limiting volume before solving. Then solve each initial-value problem and check the predictions. State the units of 6 and 0.2, and explain why changing the initial volume changes the particular solution without changing the balance law.

Related interactive example

Explore this related example, then return to the prediction task above.

The family y = ln(x² + 1) + C

Each dash shows the slope the differential equation gives at that point. Change the starting value and the solution follows the dashes.

Starting value of y at x = 00
Read the solution at x2
Try an example
y at x = 2
1.609
Rate dy/dx there
0.8
  • Solution that starts at the purple dot
  • Closed-form answer
Key idea

All solutions are copies of ln⁡(1+x2)\ln(1+x^2) shifted vertically. The particular one through the origin has constant C=0C=0.

Ideas to try
  1. Move the starting value up and down: the curves are vertical shifts of each other.
  2. Read the gradient at x=1x=1 and at x=4x=4.
Open the full interactive problem →
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