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)N9 hN = ?

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.17 h⁻¹. How many cells N are there after 9 h?

N0 = 1500k = 0.17 h⁻¹t = 9 h
N =
2.Easy
t (days)m (mg)15 dm = ?

A radioactive tracer has a half-life of 6 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 15 days?

m0 = 120 mgt½ = 6 dt = 15 d
m = mg
3.Medium
tyy(0) = 4y(1.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 4 and y′(0) = 0. Find y(1.5).

y″ + 8y′ + 15y = 0y(0) = 4y′(0) = 0
y(1.5) =
4.Medium
t (min)T (°C)22Ta​17 °C19 °C

A can of soda at 6 °C is taken out of the fridge into a room at 23 °C. After 22 min its temperature is 17 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 19 °C?

T0 = 6 °CTa = 23 °CT(22) = 17 °CT = 19 °C
t = min
5.Hard
tyK = 10y(1.5) = ?

A system obeys y″ + 4y′ + 29y = 290 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 4y′ + 29y = 290y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
xy13(1, 3)y(3) = ?

Solve the initial value problem y′ + y/x = 2x2 for x > 0, with y(1) = 3. Find y(3).

y′ + y/x = 2x2y(1) = 3
y(3) =

Answer key

  1. 1. N = 6927
  2. 2. m = 21.21 mg
  3. 3. y(1.5) = 0.1078
  4. 4. t = 30.57 min
  5. 5. y(1.5) = 9.641
  6. 6. y(3) = 14.33

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