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 (days)m (mg)27 dm = ?

A radioactive tracer has a half-life of 7 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 27 days?

m0 = 100 mgt½ = 7 dt = 27 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 8y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 12y = 0
r =
3.Medium
t (min)T (°C)26Ta​T = ?

A forged steel part at 320 °C is left to cool in a workshop at 21 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 26 min.

T0 = 320 °CTa = 21 °Ck = 0.05 min⁻¹t = 26 min
T = °C
4.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 1 and y′(0) = 1. Find y(1.5).

y″ + 4y′ + 3y = 0y(0) = 1y′(0) = 1
y(1.5) =
5.Hard
ty2y(2) = ?

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

y′ + 0.5y = 3e−3ty(0) = 2
y(2) =
6.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 2y′ + 26y = 130 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 26y = 130y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. m = 6.901 mg
  2. 2. r = −2
  3. 3. T = 102.5 °C
  4. 4. y(1.5) = 0.4352
  5. 5. y(2) = 1.174
  6. 6. ymax = 7.667

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