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 (h)Nt = ?10000

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.24 h⁻¹. After how many hours t are there 10000 cells?

N0 = 2000k = 0.24 h⁻¹N = 10000
t = h
3.Medium
tyy(0) = 1y(1) = ?

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

y″ + 2y′ + y = 0y(0) = 1y′(0) = 1
y(1) =
4.Medium
t (min)T (°C)23Ta​29 °C72 °C

A cup of coffee at 91 °C is left in a room at 24 °C. After 23 min its temperature is 29 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 72 °C?

T0 = 91 °CTa = 24 °CT(23) = 29 °CT = 72 °C
t = min
5.Hard
xy12(1, 5)y(2) = ?

Solve the initial value problem y′ + y/x = 3 for x > 0, with y(1) = 5. Find y(2).

y′ + y/x = 3y(1) = 5
y(2) =
6.Hard
t (min)m (kg)10 minm = ?

A tank holds 100 L of water in which 5 kg of salt is dissolved. Brine with 0.2 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 10 min?

V = 100 Lm0 = 5 kgc = 0.2 kg/Lr = 10 L/min
m = kg

Answer key

  1. 1. r = −1
  2. 2. t = 6.706 h
  3. 3. y(1) = 1.104
  4. 4. t = 2.956 min
  5. 5. y(2) = 4.75
  6. 6. m = 14.48 kg

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