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″ + 5y′ + 4y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 4y = 0
r =
2.Easy
t (h)N14 hN = ?

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.25 h⁻¹. How many cells N are there after 14 h?

N0 = 200k = 0.25 h⁻¹t = 14 h
N =
3.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 1 and y′(0) = 0. Find y(1.5).

y″ + 4y′ + 4y = 0y(0) = 1y′(0) = 0
y(1.5) =
4.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 1 and y′(0) = 1. Find y(1.5).

y″ + 6y′ + 5y = 0y(0) = 1y′(0) = 1
y(1.5) =
5.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) =
6.Hard
t (min)m (kg)t = ?25 kg

A tank holds 1000 L of pure water. Brine with 0.05 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 25 kg of salt?

V = 1000 Lm0 = 0 kgc = 0.05 kg/Lr = 5 L/min
t = min

Answer key

  1. 1. r = −1
  2. 2. N = 6623
  3. 3. y(1.5) = 0.1991
  4. 4. y(1.5) = 0.3344
  5. 5. y(0.25) = 3.62
  6. 6. t = 138.6 min

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