Differential equations
Order and degree, and simple solutions.
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A differential equation describes how something changes.
Look: Growth that depends on size: slow at first, then faster and faster.
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1500 doubles every hour. After 4 hours?
2Order of d²y/dx² + y = 0?
3Solve dy/dx = 2x
! Careful!
✗Thinking growth adds the same each hour.
✓It multiplies: it gets faster.
✗Forgetting the starting amount.
✓The start (1,000) fixes which curve you are on.
See the whole lesson on one page
A differential equation describes how something changes.
If something grows at a rate that matches its size, it doubles again and again in equal time. That is exponential growth.
Think of it like thisBacteria: the more there are, the faster they grow. "Growth depends on size" is a differential equation.
Solved together
1,000 bacteria double every 3 hours. How many after 9 hours?
9 hours is 3 doubling times.
9 ÷ 3 = 3Double three times.
1,000 → 2,000 → 4,000 → 8,000
Answer8,000 bacteria.
For older learners and grown-ups: a little more
dN/dt = kN separates to dN/N = k dt, so ln N = kt + C and N = N₀e^(kt). With N(3) = 2N₀, e^(3k) = 2, so N(9) = N₀ × 2³.
Now practise it
Kanakku makes new questions on differential equations every time. It shows the working and gives a hint when you are stuck.