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)21 dm = ?

A radioactive tracer has a half-life of 13 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 21 days?

m0 = 20 mgt½ = 13 dt = 21 d
m = mg
2.Easy
ty1

A mass–spring–damper is described by y″ + 4y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4y′ + 25y = 0
ζ =
3.Medium
t (min)T (°C)59Ta​56 °C43 °C

A forged steel part at 320 °C is left to cool in a workshop at 28 °C. After 59 min its temperature is 56 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 43 °C?

T0 = 320 °CTa = 28 °CT(59) = 56 °CT = 43 °C
t = min
4.Medium
t (min)T (°C)8Ta​16 °C

A can of soda at 5 °C is taken out of the fridge into a room at 29 °C. After 8 min its temperature is 16 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 5 °CTa = 29 °CT(8) = 16 °C
k = min⁻¹
5.Hard
tyy(0) = 2y(1) = ?

Solve y″ + 3y′ + 6.25y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 3y′ + 6.25y = 0y(0) = 2y′(0) = 0
y(1) =
6.Hard
ty1.5y(1.5) = ?

Solve y′ + 0.5y = 4e−2t with y(0) = 5. Find y(1.5).

y′ + 0.5y = 4e−2ty(0) = 5
y(1.5) =

Answer key

  1. 1. m = 6.528 mg
  2. 2. ζ = 0.4
  3. 3. t = 74.71 min
  4. 4. k = 0.07664 min⁻¹
  5. 5. y(1) = 0.1186
  6. 6. y(1.5) = 3.489

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