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)Nt = ?5000

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.07 h⁻¹. After how many hours t are there 5000 cells?

N0 = 1000k = 0.07 h⁻¹N = 5000
t = h
2.Easy
rr² + br + c?

For the equation y″ + 8y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 12y = 0
r =
3.Medium
t (min)T (°C)8Ta​50 °C

A cup of coffee at 93 °C is left in a room at 20 °C. After 8 min its temperature is 50 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 93 °CTa = 20 °CT(8) = 50 °C
k = min⁻¹
4.Medium
t (min)T (°C)12Ta​T = ?

A cup of coffee at 95 °C is left in a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.11 min⁻¹, find its temperature T after 12 min.

T0 = 95 °CTa = 22 °Ck = 0.11 min⁻¹t = 12 min
T = °C
5.Hard
xy13(1, 1)y(3) = ?

Solve the initial value problem y′ + 2y/x = 4 for x > 0, with y(1) = 1. Find y(3).

y′ + 2y/x = 4y(1) = 1
y(3) =
6.Hard
ty1y(1) = ?

Solve y′ + 2y = 3e−3t with y(0) = 2. Find y(1).

y′ + 2y = 3e−3ty(0) = 2
y(1) =

Answer key

  1. 1. t = 22.99 h
  2. 2. r = −2
  3. 3. k = 0.1112 min⁻¹
  4. 4. T = 41.5 °C
  5. 5. y(3) = 3.963
  6. 6. y(1) = 0.5273

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