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)y3 h224

A quantity grows according to dy/dt = k·y with k > 0. It is 80 at t = 0 and 224 at t = 3 h. Find the rate constant k in h⁻¹.

y(0) = 80y(3) = 224
k = h⁻¹
2.Easy
t (days)m (mg)68 dm = ?

A radioactive tracer has a half-life of 18 days. A sample initially contains 50 mg. The decay follows dm/dt = −k·m. How much m remains after 68 days?

m0 = 50 mgt½ = 18 dt = 68 d
m = mg
3.Medium
t (min)T (°C)23Ta​38 °C30 °C

A cup of coffee at 85 °C is left in a room at 22 °C. After 23 min its temperature is 38 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 30 °C?

T0 = 85 °CTa = 22 °CT(23) = 38 °CT = 30 °C
t = min
4.Medium
t (min)T (°C)20Ta​T = ?

A cup of coffee at 81 °C is left in a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 20 min.

T0 = 81 °CTa = 22 °Ck = 0.04 min⁻¹t = 20 min
T = °C
5.Hard
tyy(0) = 2y(1.5) = ?

Solve y″ + 2y′ + 17y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 2y′ + 17y = 0y(0) = 2y′(0) = 0
y(1.5) =
6.Hard
tyK = 1y(1) = ?

A system obeys y″ + y′ + 25.25y = 25.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + y′ + 25.25y = 25.25y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. k = 0.3432 h⁻¹
  2. 2. m = 3.645 mg
  3. 3. t = 34.63 min
  4. 4. T = 48.51 °C
  5. 5. y(1.5) = 0.3973
  6. 6. y(1) = 0.8861

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