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)N21 hN = ?

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.34 h⁻¹. How many cells N are there after 21 h?

N0 = 1500k = 0.34 h⁻¹t = 21 h
N =
2.Easy
t (h)y4 h250

A quantity grows according to dy/dt = k·y with k > 0. It is 100 at t = 0 and 250 at t = 4 h. Find the rate constant k in h⁻¹.

y(0) = 100y(4) = 250
k = h⁻¹
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 5 and y′(0) = 4. Find y(0.5).

y″ + 8y′ + 16y = 0y(0) = 5y′(0) = 4
y(0.5) =
4.Medium
tyy(0) = 3y(1) = ?

Solve y″ + 5y′ + 4y = 0 with y(0) = 3 and y′(0) = 0. Find y(1).

y″ + 5y′ + 4y = 0y(0) = 3y′(0) = 0
y(1) =
5.Hard
t (min)m (kg)t = ?34 kg

A tank holds 200 L of water in which 10 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 34 kg of salt?

V = 200 Lm0 = 10 kgc = 0.25 kg/Lr = 2 L/min
t = min
6.Hard
tyy(0) = 1y(1.5) = ?

Solve y″ + 3y′ + 6.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 3y′ + 6.25y = 0y(0) = 1y′(0) = 0
y(1.5) =

Answer key

  1. 1. N = 1.892 × 10⁶
  2. 2. k = 0.2291 h⁻¹
  3. 3. y(0.5) = 2.301
  4. 4. y(1) = 1.453
  5. 5. t = 91.63 min
  6. 6. y(1.5) = −0.09319

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