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″ + 8y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 12y = 0
r =
2.Easy
t (h)Nt = ?4500

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.24 h⁻¹. After how many hours t are there 4500 cells?

N0 = 1500k = 0.24 h⁻¹N = 4500
t = h
3.Medium
tyy(0) = 4y(2) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 4 and y′(0) = 1. Find y(2).

y″ + 8y′ + 15y = 0y(0) = 4y′(0) = 1
y(2) =
4.Medium
t (min)T (°C)32Ta​115 °C

A forged steel part at 360 °C is left to cool in a workshop at 21 °C. After 32 min its temperature is 115 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 360 °CTa = 21 °CT(32) = 115 °C
k = min⁻¹
5.Hard
tyy(0) = 1y(1.5) = ?

Solve y″ + 3y′ + 11.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 3y′ + 11.25y = 0y(0) = 1y′(0) = 0
y(1.5) =
6.Hard
tyy(0) = 1y(0.5) = ?

Solve y″ + y′ + 9.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + y′ + 9.25y = 0y(0) = 1y′(0) = 0
y(0.5) =

Answer key

  1. 1. r = −2
  2. 2. t = 4.578 h
  3. 3. y(2) = 0.02573
  4. 4. k = 0.04008 min⁻¹
  5. 5. y(1.5) = −0.07373
  6. 6. y(0.5) = 0.1846

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