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)60 dm = ?

A radioactive tracer has a half-life of 19 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 60 days?

m0 = 200 mgt½ = 19 dt = 60 d
m = mg
2.Easy
ty1

A mass–spring–damper is described by y″ + 5y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 5y′ + 25y = 0
ζ =
3.Medium
t (min)T (°C)27Ta​27 °C

A can of soda at 6 °C is taken out of the fridge into a room at 30 °C. After 27 min its temperature is 27 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 6 °CTa = 30 °CT(27) = 27 °C
k = min⁻¹
4.Medium
tyy(0) = 3y(1) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 3 and y′(0) = 1. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 3y′(0) = 1
y(1) =
5.Hard
t (min)m (kg)40 minm = ?

A tank holds 500 L of water in which 5 kg of salt is dissolved. Brine with 0.05 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 40 min?

V = 500 Lm0 = 5 kgc = 0.05 kg/Lr = 10 L/min
m = kg
6.Hard
xy12(1, 2)y(2) = ?

Solve the initial value problem y′ + 3y/x = 5 for x > 0, with y(1) = 2. Find y(2).

y′ + 3y/x = 5y(1) = 2
y(2) =

Answer key

  1. 1. m = 22.41 mg
  2. 2. ζ = 0.5
  3. 3. k = 0.07702 min⁻¹
  4. 4. y(1) = 0.6472
  5. 5. m = 16.01 kg
  6. 6. y(2) = 2.594

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