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″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 4y′ + 3y = 0
r =
2.Easy
t (h)y9 h125

A quantity decays according to dy/dt = −k·y with k > 0. It is 250 at t = 0 and 125 at t = 9 h. Find the rate constant k in h⁻¹.

y(0) = 250y(9) = 125
k = h⁻¹
3.Medium
t (min)T (°C)43Ta​99 °C66 °C

A forged steel part at 440 °C is left to cool in a workshop at 25 °C. After 43 min its temperature is 99 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 66 °C?

T0 = 440 °CTa = 25 °CT(43) = 99 °CT = 66 °C
t = min
4.Medium
t (min)T (°C)20Ta​T = ?

A cup of coffee at 81 °C is left in a room at 20 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.12 min⁻¹, find its temperature T after 20 min.

T0 = 81 °CTa = 20 °Ck = 0.12 min⁻¹t = 20 min
T = °C
5.Hard
tyy(0) = 3y(0.25) = ?

Solve y″ + 4y′ + 13y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 4y′ + 13y = 0y(0) = 3y′(0) = 0
y(0.25) =
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + y′ + 9.25y = 92.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 9.25y = 92.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. r = −1
  2. 2. k = 0.07702 h⁻¹
  3. 3. t = 57.73 min
  4. 4. T = 25.53 °C
  5. 5. y(0.25) = 2.158
  6. 6. ymax = 15.92

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