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 (days)m (mg)26 dm = ?

A radioactive tracer has a half-life of 11 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 26 days?

m0 = 20 mgt½ = 11 dt = 26 d
m = mg
2.Easy
t (h)y6 h90

A quantity decays according to dy/dt = −k·y with k > 0. It is 120 at t = 0 and 90 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 120y(6) = 90
k = h⁻¹
3.Medium
t (min)T (°C)6Ta​11 °C

A can of soda at 3 °C is taken out of the fridge into a room at 25 °C. After 6 min its temperature is 11 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 3 °CTa = 25 °CT(6) = 11 °C
k = min⁻¹
4.Medium
tyy(0) = 1y(2) = ?

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

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

Solve y″ + 4y′ + 8y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 2y′(0) = 0
y(1) =
6.Hard
xy13(1, −3)y(3) = ?

Solve the initial value problem y′ + y/x = 3 for x > 0, with y(1) = −3. Find y(3).

y′ + y/x = 3y(1) = −3
y(3) =

Answer key

  1. 1. m = 3.886 mg
  2. 2. k = 0.04795 h⁻¹
  3. 3. k = 0.07533 min⁻¹
  4. 4. y(2) = 0.09158
  5. 5. y(1) = 0.1335
  6. 6. y(3) = 3

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