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)7 dm = ?

A radioactive tracer has a half-life of 20 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 7 days?

m0 = 20 mgt½ = 20 dt = 7 d
m = mg
2.Easy
ty1

A mass–spring–damper is described by y″ + 0.4y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 0.4y′ + 4y = 0
ζ =
3.Medium
t (min)T (°C)34Ta​349 °C415 °C

A forged steel part at 480 °C is left to cool in a workshop at 25 °C. After 34 min its temperature is 349 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 415 °C?

T0 = 480 °CTa = 25 °CT(34) = 349 °CT = 415 °C
t = min
4.Medium
tyy(0) = 4y(1) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 4 and y′(0) = 2. Find y(1).

y″ + 8y′ + 15y = 0y(0) = 4y′(0) = 2
y(1) =
5.Hard
tyK = 2y(0.5) = ?

A system obeys y″ + 2y′ + 26y = 52 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 2y′ + 26y = 52y(0) = 0y′(0) = 0
y(0.5) =
6.Hard
xy13(1, −3)y(3) = ?

Solve the initial value problem y′ + 2y/x = 6x2 for x > 0, with y(1) = −3. Find y(3).

y′ + 2y/x = 6x2y(1) = −3
y(3) =

Answer key

  1. 1. m = 15.69 mg
  2. 2. ζ = 0.1
  3. 3. t = 15.44 min
  4. 4. y(1) = 0.5005
  5. 5. y(0.5) = 2.827
  6. 6. y(3) = 31.93

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