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 21 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 8 days?

m0 = 20 mgt½ = 21 dt = 8 d
m = mg
2.Easy
t (h)y7 h50

A quantity decays according to dy/dt = −k·y with k > 0. It is 100 at t = 0 and 50 at t = 7 h. Find the rate constant k in h⁻¹.

y(0) = 100y(7) = 50
k = h⁻¹
3.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 5 and y′(0) = 1. Find y(2).

y″ + 8y′ + 16y = 0y(0) = 5y′(0) = 1
y(2) =
4.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 5 and y′(0) = -2. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 5y′(0) = -2
y(0.5) =
5.Hard
xy13(1, 5)y(3) = ?

Solve the initial value problem y′ + 3y/x = 3 for x > 0, with y(1) = 5. Find y(3).

y′ + 3y/x = 3y(1) = 5
y(3) =
6.Hard
xy13(1, −1)y(3) = ?

Solve the initial value problem y′ + y/x = 5x2 for x > 0, with y(1) = −1. Find y(3).

y′ + y/x = 5x2y(1) = −1
y(3) =

Answer key

  1. 1. m = 15.36 mg
  2. 2. k = 0.09902 h⁻¹
  3. 3. y(2) = 0.01577
  4. 4. y(0.5) = 3.311
  5. 5. y(3) = 2.407
  6. 6. y(3) = 33

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