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″ + 5y′ + 6y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 6y = 0
r =
2.Easy
t (days)m (mg)30 dm = ?

A radioactive tracer has a half-life of 16 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 30 days?

m0 = 10 mgt½ = 16 dt = 30 d
m = mg
3.Medium
t (min)T (°C)34Ta​263 °C166 °C

A forged steel part at 500 °C is left to cool in a workshop at 20 °C. After 34 min its temperature is 263 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 166 °C?

T0 = 500 °CTa = 20 °CT(34) = 263 °CT = 166 °C
t = min
4.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 2 and y′(0) = 4. Find y(1.5).

y″ + 8y′ + 15y = 0y(0) = 2y′(0) = 4
y(1.5) =
5.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 2y′ + 10y = 100 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 10y = 100y(0) = 0y′(0) = 0
ymax =
6.Hard
tyy(0) = 1y(0.75) = ?

Solve y″ + 2y′ + 10y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 2y′ + 10y = 0y(0) = 1y′(0) = 0
y(0.75) =

Answer key

  1. 1. r = −2
  2. 2. m = 2.726 mg
  3. 3. t = 59.45 min
  4. 4. y(1.5) = 0.075
  5. 5. ymax = 13.51
  6. 6. y(0.75) = −0.1742

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