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)y9 h43

A quantity decays according to dy/dt = −k·y with k > 0. It is 50 at t = 0 and 43 at t = 9 h. Find the rate constant k in h⁻¹.

y(0) = 50y(9) = 43
k = h⁻¹
2.Easy
t (days)m (mg)67 dm = ?

A radioactive tracer has a half-life of 23 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 67 days?

m0 = 10 mgt½ = 23 dt = 67 d
m = mg
3.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = -1. Find y(1.5).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = -1
y(1.5) =
4.Medium
tyy(0) = 5y(2) = ?

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

y″ + 6y′ + 5y = 0y(0) = 5y′(0) = 0
y(2) =
5.Hard
tyK = 5y(1) = ?

A system obeys y″ + 4y′ + 29y = 145 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 29y = 145y(0) = 0y′(0) = 0
y(1) =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 4y′ + 20y = 40 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

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

Answer key

  1. 1. k = 0.01676 h⁻¹
  2. 2. m = 1.328 mg
  3. 3. y(1.5) = 0.2888
  4. 4. y(2) = 0.8458
  5. 5. y(1) = 5.068
  6. 6. ymax = 2.416

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