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 (h)y3 h50

A quantity decays according to dy/dt = −k·y with k > 0. It is 250 at t = 0 and 50 at t = 3 h. Find the rate constant k in h⁻¹.

y(0) = 250y(3) = 50
k = h⁻¹
2.Easy
rr² + br + c?

For the equation y″ + 9y′ + 18y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 9y′ + 18y = 0
r =
3.Medium
t (min)T (°C)43Ta​231 °C286 °C

A forged steel part at 340 °C is left to cool in a workshop at 27 °C. After 43 min its temperature is 231 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 286 °C?

T0 = 340 °CTa = 27 °CT(43) = 231 °CT = 286 °C
t = min
4.Medium
tyy(0) = 2y(2) = ?

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

y″ + 4y′ + 4y = 0y(0) = 2y′(0) = 2
y(2) =
5.Hard
t (min)m (kg)30 minm = ?

A tank holds 100 L of water in which 20 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 30 min?

V = 100 Lm0 = 20 kgc = 0.25 kg/Lr = 5 L/min
m = kg
6.Hard
tyy(0) = 10y(0.25) = ?

Solve y″ + y′ + 25.25y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + y′ + 25.25y = 0y(0) = 10y′(0) = 0
y(0.25) =

Answer key

  1. 1. k = 0.5365 h⁻¹
  2. 2. r = −3
  3. 3. t = 19.02 min
  4. 4. y(2) = 0.2564
  5. 5. m = 23.88 kg
  6. 6. y(0.25) = 3.62

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