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″ + 8y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8y′ + 25y = 0
ζ =
2.Easy
t (days)m (mg)35 dm = ?

A radioactive tracer has a half-life of 11 days. A sample initially contains 40 mg. The decay follows dm/dt = −k·m. How much m remains after 35 days?

m0 = 40 mgt½ = 11 dt = 35 d
m = mg
3.Medium
t (min)T (°C)25Ta​53 °C

A cup of coffee at 87 °C is left in a room at 23 °C. After 25 min its temperature is 53 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 87 °CTa = 23 °CT(25) = 53 °C
k = min⁻¹
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 5 and y′(0) = 1. Find y(2).

y″ + 2y′ + y = 0y(0) = 5y′(0) = 1
y(2) =
5.Hard
ty2y(2) = ?

Solve y′ + y = 10e−3t with y(0) = 4. Find y(2).

y′ + y = 10e−3ty(0) = 4
y(2) =
6.Hard
t (min)m (kg)40 minm = ?

A tank holds 400 L of pure water. 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. How much salt m is in the tank after 40 min?

V = 400 Lm0 = 0 kgc = 0.2 kg/Lr = 2 L/min
m = kg

Answer key

  1. 1. ζ = 0.8
  2. 2. m = 4.408 mg
  3. 3. k = 0.03031 min⁻¹
  4. 4. y(2) = 2.301
  5. 5. y(2) = 1.206
  6. 6. m = 14.5 kg

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