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)y6 h1320

A quantity grows according to dy/dt = k·y with k > 0. It is 400 at t = 0 and 1320 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 400y(6) = 1320
k = h⁻¹
2.Easy
t (h)y8 h300

A quantity decays according to dy/dt = −k·y with k > 0. It is 500 at t = 0 and 300 at t = 8 h. Find the rate constant k in h⁻¹.

y(0) = 500y(8) = 300
k = h⁻¹
3.Medium
t (min)T (°C)20Ta​185 °C70 °C

A forged steel part at 320 °C is left to cool in a workshop at 20 °C. After 20 min its temperature is 185 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 70 °C?

T0 = 320 °CTa = 20 °CT(20) = 185 °CT = 70 °C
t = min
4.Medium
t (min)T (°C)45Ta​87 °C

A forged steel part at 420 °C is left to cool in a workshop at 21 °C. After 45 min its temperature is 87 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 420 °CTa = 21 °CT(45) = 87 °C
k = min⁻¹
5.Hard
xy13(1, 1)y(3) = ?

Solve the initial value problem y′ + 3y/x = 5x for x > 0, with y(1) = 1. Find y(3).

y′ + 3y/x = 5xy(1) = 1
y(3) =
6.Hard
xy12(1, 2)y(2) = ?

Solve the initial value problem y′ + 2y/x = 5 for x > 0, with y(1) = 2. Find y(2).

y′ + 2y/x = 5y(1) = 2
y(2) =

Answer key

  1. 1. k = 0.199 h⁻¹
  2. 2. k = 0.06385 h⁻¹
  3. 3. t = 59.94 min
  4. 4. k = 0.03998 min⁻¹
  5. 5. y(3) = 9
  6. 6. y(2) = 3.417

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