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)N5 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.18 h⁻¹. How many cells N are there after 5 h?

N0 = 2000k = 0.18 h⁻¹t = 5 h
N =
2.Easy
rr² + br + c?

For the equation y″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 3y′ + 2y = 0
r =
3.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 2 and y′(0) = 2. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 2y′(0) = 2
y(1.5) =
4.Medium
t (min)T (°C)46Ta​191 °C365 °C

A forged steel part at 440 °C is left to cool in a workshop at 26 °C. After 46 min its temperature is 191 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 365 °C?

T0 = 440 °CTa = 26 °CT(46) = 191 °CT = 365 °C
t = min
5.Hard
tyy(0) = 10y(1) = ?

Solve y″ + 3y′ + 6.25y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 3y′ + 6.25y = 0y(0) = 10y′(0) = 0
y(1) =
6.Hard
tyy(0) = 10y(1) = ?

Solve y″ + 4y′ + 13y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 4y′ + 13y = 0y(0) = 10y′(0) = 0
y(1) =

Answer key

  1. 1. N = 4919
  2. 2. r = −1
  3. 3. y(1.5) = 1.785
  4. 4. t = 9.994 min
  5. 5. y(1) = 0.5931
  6. 6. y(1) = −1.212

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