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)N6 hN = ?

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.29 h⁻¹. How many cells N are there after 6 h?

N0 = 1500k = 0.29 h⁻¹t = 6 h
N =
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
tyy(0) = 3y(1) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 3 and y′(0) = -4. Find y(1).

y″ + 7y′ + 10y = 0y(0) = 3y′(0) = -4
y(1) =
4.Medium
t (min)T (°C)11Ta​57 °C

A cup of coffee at 82 °C is left in a room at 22 °C. After 11 min its temperature is 57 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 82 °CTa = 22 °CT(11) = 57 °C
k = min⁻¹
5.Hard
tyK = 5y(0.75) = ?

A system obeys y″ + 4y′ + 29y = 145 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 4y′ + 29y = 145y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
tyK = 10y(1) = ?

A system obeys y″ + 3y′ + 6.25y = 62.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 3y′ + 6.25y = 62.5y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. N = 8546
  2. 2. r = −3
  3. 3. y(1) = 0.4917
  4. 4. k = 0.049 min⁻¹
  5. 5. y(0.75) = 6.171
  6. 6. y(1) = 9.407

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