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 (h)y10 h65

A quantity grows according to dy/dt = k·y with k > 0. It is 50 at t = 0 and 65 at t = 10 h. Find the rate constant k in h⁻¹.

y(0) = 50y(10) = 65
k = h⁻¹
2.Easy
t (days)m (mg)7 dm = ?

A radioactive tracer has a half-life of 23 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 7 days?

m0 = 120 mgt½ = 23 dt = 7 d
m = mg
3.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 2 and y′(0) = 4. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 2y′(0) = 4
y(1) =
4.Medium
tyy(0) = 1y(2) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 1 and y′(0) = 0. Find y(2).

y″ + 6y′ + 8y = 0y(0) = 1y′(0) = 0
y(2) =
5.Hard
tyy(0) = 3y(0.75) = ?

Solve y″ + y′ + 4.25y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + y′ + 4.25y = 0y(0) = 3y′(0) = 0
y(0.75) =
6.Hard
tyy(0) = 10y(1.5) = ?

Solve y″ + y′ + 9.25y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + y′ + 9.25y = 0y(0) = 10y′(0) = 0
y(1.5) =

Answer key

  1. 1. k = 0.02624 h⁻¹
  2. 2. m = 97.18 mg
  3. 3. y(1) = 0.5974
  4. 4. y(2) = 0.0363
  5. 5. y(0.75) = 0.66
  6. 6. y(1.5) = −1.765

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