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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
t (h)y9 h775

A quantity grows according to dy/dt = k·y with k > 0. It is 250 at t = 0 and 775 at t = 9 h. Find the rate constant k in h⁻¹.

y(0) = 250y(9) = 775
k = h⁻¹
2.Easy
t (days)m (mg)59 dm = ?

A radioactive tracer has a half-life of 22 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 59 days?

m0 = 100 mgt½ = 22 dt = 59 d
m = mg
3.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 1 and y′(0) = -2. Find y(0.5).

y″ + 6y′ + 9y = 0y(0) = 1y′(0) = -2
y(0.5) =
4.Medium
t (min)T (°C)44Ta​203 °C89 °C

A forged steel part at 300 °C is left to cool in a workshop at 28 °C. After 44 min its temperature is 203 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 89 °C?

T0 = 300 °CTa = 28 °CT(44) = 203 °CT = 89 °C
t = min
5.Hard
ty2y(2) = ?

Solve y′ + 3y = 5e−0.5t with y(0) = 2. Find y(2).

y′ + 3y = 5e−0.5ty(0) = 2
y(2) =
6.Hard
tyy(0) = 1y(1.5) = ?

Solve y″ + 4y′ + 8y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 1y′(0) = 0
y(1.5) =

Answer key

  1. 1. k = 0.1257 h⁻¹
  2. 2. m = 15.58 mg
  3. 3. y(0.5) = 0.3347
  4. 4. t = 149.1 min
  5. 5. y(2) = 0.7358
  6. 6. y(1.5) = −0.04226

Full step-by-step solutions: practise this topic on examproblems.com.