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 (days)m (mg)12 dm = ?

A radioactive tracer has a half-life of 15 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 12 days?

m0 = 120 mgt½ = 15 dt = 12 d
m = mg
2.Easy
t (days)m (mg)54 dm = ?

A radioactive tracer has a half-life of 23 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 54 days?

m0 = 100 mgt½ = 23 dt = 54 d
m = mg
3.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 5y′ + 4y = 0 with y(0) = 1 and y′(0) = 6. Find y(1).

y″ + 5y′ + 4y = 0y(0) = 1y′(0) = 6
y(1) =
4.Medium
t (min)T (°C)16Ta​T = ?

A forged steel part at 420 °C is left to cool in a workshop at 20 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.01 min⁻¹, find its temperature T after 16 min.

T0 = 420 °CTa = 20 °Ck = 0.01 min⁻¹t = 16 min
T = °C
5.Hard
ty0.5y(0.5) = ?

Solve y′ + 0.5y = 3e−t with y(0) = 3. Find y(0.5).

y′ + 0.5y = 3e−ty(0) = 3
y(0.5) =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + y′ + 25.25y = 50.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 25.25y = 50.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. m = 68.92 mg
  2. 2. m = 19.64 mg
  3. 3. y(1) = 1.184
  4. 4. T = 360.9 °C
  5. 5. y(0.5) = 3.37
  6. 6. ymax = 3.461

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