Differential equations worksheet with answers

Free printable Differential equations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
t (h)N6 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.05 h⁻¹. How many cells N are there after 6 h?

N0 = 500k = 0.05 h⁻¹t = 6 h
N =
2.Easy
t (h)y11 h40

A quantity decays according to dy/dt = −k·y with k > 0. It is 50 at t = 0 and 40 at t = 11 h. Find the rate constant k in h⁻¹.

y(0) = 50y(11) = 40
k = h⁻¹
3.Medium
t (min)T (°C)12Ta​42 °C54 °C

A cup of coffee at 81 °C is left in a room at 22 °C. After 12 min its temperature is 42 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 54 °C?

T0 = 81 °CTa = 22 °CT(12) = 42 °CT = 54 °C
t = min
4.Medium
tyy(0) = 2y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 2 and y′(0) = -1. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 2y′(0) = -1
y(2) =
5.Hard
tyy(0) = 10y(0.5) = ?

Solve y″ + 2y′ + 5y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 2y′ + 5y = 0y(0) = 10y′(0) = 0
y(0.5) =
6.Hard
xy12(1, 2)y(2) = ?

Solve the initial value problem y′ + y/x = 5x2 for x > 0, with y(1) = 2. Find y(2).

y′ + y/x = 5x2y(1) = 2
y(2) =

Answer key

  1. 1. N = 674.9
  2. 2. k = 0.02029 h⁻¹
  3. 3. t = 6.786 min
  4. 4. y(2) = 0.1465
  5. 5. y(0.5) = 5.829
  6. 6. y(2) = 10.38

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