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″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 4y′ + 3y = 0
r =
2.Easy
t (h)Nt = ?5000

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.05 h⁻¹. After how many hours t are there 5000 cells?

N0 = 500k = 0.05 h⁻¹N = 5000
t = h
3.Medium
t (min)T (°C)20Ta​28 °C

A cup of coffee at 93 °C is left in a room at 21 °C. After 20 min its temperature is 28 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 93 °CTa = 21 °CT(20) = 28 °C
k = min⁻¹
4.Medium
t (min)T (°C)30Ta​T = ?

A forged steel part at 400 °C is left to cool in a workshop at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 30 min.

T0 = 400 °CTa = 22 °Ck = 0.03 min⁻¹t = 30 min
T = °C
5.Hard
ty1.5y(1.5) = ?

Solve y′ + y = 3e−3t with y(0) = 4. Find y(1.5).

y′ + y = 3e−3ty(0) = 4
y(1.5) =
6.Hard
tyK = 10y(1.5) = ?

A system obeys y″ + y′ + 25.25y = 252.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + y′ + 25.25y = 252.5y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. r = −1
  2. 2. t = 46.05 h
  3. 3. k = 0.1165 min⁻¹
  4. 4. T = 175.7 °C
  5. 5. y(1.5) = 1.211
  6. 6. y(1.5) = 7.92

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