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)32 dm = ?

A radioactive tracer has a half-life of 26 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 32 days?

m0 = 10 mgt½ = 26 dt = 32 d
m = mg
2.Easy
t (h)y6 h1080

A quantity grows according to dy/dt = k·y with k > 0. It is 400 at t = 0 and 1080 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 400y(6) = 1080
k = h⁻¹
3.Medium
t (min)T (°C)6Ta​67 °C

A cup of coffee at 94 °C is left in a room at 24 °C. After 6 min its temperature is 67 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 94 °CTa = 24 °CT(6) = 67 °C
k = min⁻¹
4.Medium
tyy(0) = 3y(1) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 3 and y′(0) = 1. Find y(1).

y″ + 7y′ + 12y = 0y(0) = 3y′(0) = 1
y(1) =
5.Hard
xy12(1, −3)y(2) = ?

Solve the initial value problem y′ + 3y/x = 5 for x > 0, with y(1) = −3. Find y(2).

y′ + 3y/x = 5y(1) = −3
y(2) =
6.Hard
tyy(0) = 10y(1.5) = ?

Solve y″ + 3y′ + 18.25y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 3y′ + 18.25y = 0y(0) = 10y′(0) = 0
y(1.5) =

Answer key

  1. 1. m = 4.261 mg
  2. 2. k = 0.1655 h⁻¹
  3. 3. k = 0.08122 min⁻¹
  4. 4. y(1) = 0.4641
  5. 5. y(2) = 1.969
  6. 6. y(1.5) = 0.9016

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