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
ty1

A mass–spring–damper is described by y″ + 9.6y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 9.6y′ + 36y = 0
ζ =
2.Easy
t (days)m (mg)23 dm = ?

A radioactive tracer has a half-life of 6 days. A sample initially contains 40 mg. The decay follows dm/dt = −k·m. How much m remains after 23 days?

m0 = 40 mgt½ = 6 dt = 23 d
m = mg
3.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 2 and y′(0) = -4. Find y(1.5).

y″ + 7y′ + 12y = 0y(0) = 2y′(0) = -4
y(1.5) =
4.Medium
t (min)T (°C)15Ta​33 °C29 °C

A cup of coffee at 81 °C is left in a room at 23 °C. After 15 min its temperature is 33 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 29 °C?

T0 = 81 °CTa = 23 °CT(15) = 33 °CT = 29 °C
t = min
5.Hard
t (min)m (kg)t = ?14 kg

A tank holds 100 L of water in which 5 kg of salt is dissolved. Brine with 0.2 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 14 kg of salt?

V = 100 Lm0 = 5 kgc = 0.2 kg/Lr = 2 L/min
t = min
6.Hard
xy13(1, 5)y(3) = ?

Solve the initial value problem y′ + y/x = 6x2 for x > 0, with y(1) = 5. Find y(3).

y′ + y/x = 6x2y(1) = 5
y(3) =

Answer key

  1. 1. ζ = 0.8
  2. 2. m = 2.806 mg
  3. 3. y(1.5) = 0.03948
  4. 4. t = 19.36 min
  5. 5. t = 45.81 min
  6. 6. y(3) = 41.67

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