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″ + 6y′ + 8y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 8y = 0
r =
2.Easy
t (days)m (mg)20 dm = ?

A radioactive tracer has a half-life of 6 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 20 days?

m0 = 120 mgt½ = 6 dt = 20 d
m = mg
3.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 2 and y′(0) = -2. Find y(0.5).

y″ + 2y′ + y = 0y(0) = 2y′(0) = -2
y(0.5) =
4.Medium
t (min)T (°C)26Ta​25 °C

A can of soda at 4 °C is taken out of the fridge into a room at 30 °C. After 26 min its temperature is 25 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 4 °CTa = 30 °CT(26) = 25 °C
k = min⁻¹
5.Hard
tyy(0) = 10y(0.75) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 10y′(0) = 0
y(0.75) =
6.Hard
tyK = 10y(1.5) = ?

A system obeys y″ + 4y′ + 20y = 200 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 4y′ + 20y = 200y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. r = −2
  2. 2. m = 11.91 mg
  3. 3. y(0.5) = 1.213
  4. 4. k = 0.06341 min⁻¹
  5. 5. y(0.75) = −4.416
  6. 6. y(1.5) = 9.592

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