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)3 dm = ?

A radioactive tracer has a half-life of 8 days. A sample initially contains 50 mg. The decay follows dm/dt = −k·m. How much m remains after 3 days?

m0 = 50 mgt½ = 8 dt = 3 d
m = mg
2.Easy
t (h)N16 hN = ?

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.32 h⁻¹. How many cells N are there after 16 h?

N0 = 1000k = 0.32 h⁻¹t = 16 h
N =
3.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 3 and y′(0) = 2. Find y(0.5).

y″ + 2y′ + y = 0y(0) = 3y′(0) = 2
y(0.5) =
4.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = 4. Find y(0.5).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = 4
y(0.5) =
5.Hard
tyK = 2y(0.75) = ?

A system obeys y″ + 4y′ + 8y = 16 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 4y′ + 8y = 16y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
tyK = 5y(0.75) = ?

A system obeys y″ + 2y′ + 17y = 85 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 2y′ + 17y = 85y(0) = 0y′(0) = 0
y(0.75) =

Answer key

  1. 1. m = 38.56 mg
  2. 2. N = 167335
  3. 3. y(0.5) = 3.336
  4. 4. y(0.5) = 3.235
  5. 5. y(0.75) = 1.523
  6. 6. y(0.75) = 7.255

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