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)y3 h780

A quantity grows according to dy/dt = k·y with k > 0. It is 200 at t = 0 and 780 at t = 3 h. Find the rate constant k in h⁻¹.

y(0) = 200y(3) = 780
k = h⁻¹
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 6y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 6y = 0
r =
3.Medium
t (min)T (°C)47Ta​T = ?

A forged steel part at 460 °C is left to cool in a workshop at 27 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 47 min.

T0 = 460 °CTa = 27 °Ck = 0.05 min⁻¹t = 47 min
T = °C
4.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 2 and y′(0) = -2. Find y(1).

y″ + 7y′ + 10y = 0y(0) = 2y′(0) = -2
y(1) =
5.Hard
tyK = 2y(1) = ?

A system obeys y″ + 4y′ + 8y = 16 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 8y = 16y(0) = 0y′(0) = 0
y(1) =
6.Hard
tyK = 5y(0.75) = ?

A system obeys y″ + 3y′ + 18.25y = 91.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 3y′ + 18.25y = 91.25y(0) = 0y′(0) = 0
y(0.75) =

Answer key

  1. 1. k = 0.4537 h⁻¹
  2. 2. r = −1
  3. 3. T = 68.29 °C
  4. 4. y(1) = 0.3564
  5. 5. y(1) = 1.867
  6. 6. y(0.75) = 6.521

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