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
rr² + br + c?

For the equation y″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 3y′ + 2y = 0
r =
2.Easy
t (days)m (mg)11 dm = ?

A radioactive tracer has a half-life of 16 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 11 days?

m0 = 80 mgt½ = 16 dt = 11 d
m = mg
3.Medium
t (min)T (°C)23Ta​T = ?

A forged steel part at 360 °C is left to cool in a workshop at 24 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 23 min.

T0 = 360 °CTa = 24 °Ck = 0.04 min⁻¹t = 23 min
T = °C
4.Medium
t (min)T (°C)8Ta​T = ?

A can of soda at 5 °C is taken out of the fridge into a room at 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.1 min⁻¹, find its temperature T after 8 min.

T0 = 5 °CTa = 23 °Ck = 0.1 min⁻¹t = 8 min
T = °C
5.Hard
tyy(0) = 1y(0.25) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 1y′(0) = 0
y(0.25) =
6.Hard
tyK = 2y(0.5) = ?

A system obeys y″ + 2y′ + 17y = 34 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 2y′ + 17y = 34y(0) = 0y′(0) = 0
y(0.5) =

Answer key

  1. 1. r = −1
  2. 2. m = 49.67 mg
  3. 3. T = 157.9 °C
  4. 4. T = 14.91 °C
  5. 5. y(0.25) = 0.3934
  6. 6. y(0.5) = 2.229

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