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)8 dm = ?

A radioactive tracer has a half-life of 9 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 8 days?

m0 = 120 mgt½ = 9 dt = 8 d
m = mg
2.Easy
t (h)N22 hN = ?

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.23 h⁻¹. How many cells N are there after 22 h?

N0 = 1500k = 0.23 h⁻¹t = 22 h
N =
3.Medium
tyy(0) = 4y(2) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 4 and y′(0) = 2. Find y(2).

y″ + 6y′ + 5y = 0y(0) = 4y′(0) = 2
y(2) =
4.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 5 and y′(0) = 6. Find y(1).

y″ + 8y′ + 15y = 0y(0) = 5y′(0) = 6
y(1) =
5.Hard
xy13(1, −3)y(3) = ?

Solve the initial value problem y′ + 3y/x = 6x for x > 0, with y(1) = −3. Find y(3).

y′ + 3y/x = 6xy(1) = −3
y(3) =
6.Hard
tyK = 2y(1) = ?

A system obeys y″ + 4y′ + 20y = 40 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 20y = 40y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. m = 64.8 mg
  2. 2. N = 236386
  3. 3. y(2) = 0.7443
  4. 4. y(1) = 0.701
  5. 5. y(3) = 10.64
  6. 6. y(1) = 2.279

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