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″ + 5y′ + 4y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 4y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 1.6y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 1.6y′ + 4y = 0
ζ =
3.Medium
t (min)T (°C)23Ta​T = ?

A can of soda at 6 °C is taken out of the fridge into a room at 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.08 min⁻¹, find its temperature T after 23 min.

T0 = 6 °CTa = 23 °Ck = 0.08 min⁻¹t = 23 min
T = °C
4.Medium
t (min)T (°C)28Ta​17 °C

A can of soda at 5 °C is taken out of the fridge into a room at 23 °C. After 28 min its temperature is 17 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 5 °CTa = 23 °CT(28) = 17 °C
k = min⁻¹
5.Hard
tyK = 2y(1.5) = ?

A system obeys y″ + y′ + 4.25y = 8.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + y′ + 4.25y = 8.5y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
xy13(1, −2)y(3) = ?

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

y′ + 3y/x = x2y(1) = −2
y(3) =

Answer key

  1. 1. r = −1
  2. 2. ζ = 0.4
  3. 3. T = 20.3 °C
  4. 4. k = 0.03924 min⁻¹
  5. 5. y(1.5) = 2.902
  6. 6. y(3) = 4.42

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