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 = ?10000

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.25 h⁻¹. After how many hours t are there 10000 cells?

N0 = 5000k = 0.25 h⁻¹N = 10000
t = h
2.Easy
t (days)m (mg)30 dm = ?

A radioactive tracer has a half-life of 26 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 30 days?

m0 = 100 mgt½ = 26 dt = 30 d
m = mg
3.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 1 and y′(0) = 2. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 1y′(0) = 2
y(1) =
4.Medium
t (min)T (°C)5Ta​79 °C45 °C

A cup of coffee at 92 °C is left in a room at 18 °C. After 5 min its temperature is 79 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 45 °C?

T0 = 92 °CTa = 18 °CT(5) = 79 °CT = 45 °C
t = min
5.Hard
tyy(0) = 2y(1) = ?

Solve y″ + 4y′ + 20y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 4y′ + 20y = 0y(0) = 2y′(0) = 0
y(1) =
6.Hard
tyy(0) = 5y(1.5) = ?

Solve y″ + 4y′ + 20y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 4y′ + 20y = 0y(0) = 5y′(0) = 0
y(1.5) =

Answer key

  1. 1. t = 2.773 h
  2. 2. m = 44.94 mg
  3. 3. y(1) = 0.2987
  4. 4. t = 26.09 min
  5. 5. y(1) = −0.2793
  6. 6. y(1.5) = 0.2042

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