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)N13 hN = ?

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.22 h⁻¹. How many cells N are there after 13 h?

N0 = 1500k = 0.22 h⁻¹t = 13 h
N =
2.Easy
ty1

A mass–spring–damper is described by y″ + 6.4y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6.4y′ + 64y = 0
ζ =
3.Medium
t (min)T (°C)45Ta​T = ?

A forged steel part at 380 °C is left to cool in a workshop at 30 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 45 min.

T0 = 380 °CTa = 30 °Ck = 0.05 min⁻¹t = 45 min
T = °C
4.Medium
t (min)T (°C)44Ta​202 °C241 °C

A forged steel part at 300 °C is left to cool in a workshop at 26 °C. After 44 min its temperature is 202 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 241 °C?

T0 = 300 °CTa = 26 °CT(44) = 202 °CT = 241 °C
t = min
5.Hard
xy12(1, −3)y(2) = ?

Solve the initial value problem y′ + 2y/x = 6x for x > 0, with y(1) = −3. Find y(2).

y′ + 2y/x = 6xy(1) = −3
y(2) =
6.Hard
tyy(0) = 10y(1.5) = ?

Solve y″ + 2y′ + 5y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 2y′ + 5y = 0y(0) = 10y′(0) = 0
y(1.5) =

Answer key

  1. 1. N = 26192
  2. 2. ζ = 0.4
  3. 3. T = 66.89 °C
  4. 4. t = 24.1 min
  5. 5. y(2) = 4.875
  6. 6. y(1.5) = −2.052

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