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″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 4y′ + 3y = 0
r =
2.Easy
t (days)m (mg)8 dm = ?

A radioactive tracer has a half-life of 5 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 8 days?

m0 = 20 mgt½ = 5 dt = 8 d
m = mg
3.Medium
t (min)T (°C)27Ta​23 °C

A can of soda at 4 °C is taken out of the fridge into a room at 28 °C. After 27 min its temperature is 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 4 °CTa = 28 °CT(27) = 23 °C
k = min⁻¹
4.Medium
t (min)T (°C)23Ta​T = ?

A can of soda at 6 °C is taken out of the fridge into a room at 29 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.07 min⁻¹, find its temperature T after 23 min.

T0 = 6 °CTa = 29 °Ck = 0.07 min⁻¹t = 23 min
T = °C
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + y′ + 16.25y = 16.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 16.25y = 16.25y(0) = 0y′(0) = 0
ymax =
6.Hard
xy13(1, 2)y(3) = ?

Solve the initial value problem y′ + 3y/x = 4 for x > 0, with y(1) = 2. Find y(3).

y′ + 3y/x = 4y(1) = 2
y(3) =

Answer key

  1. 1. r = −1
  2. 2. m = 6.598 mg
  3. 3. k = 0.0581 min⁻¹
  4. 4. T = 24.4 °C
  5. 5. ymax = 1.675
  6. 6. y(3) = 3.037

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