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 worksheet Practise 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

A radioactive tracer has a half-life of 20 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 10 days?

m₀ = 20 mgt_½ = 20 dt = 10 d

m = mg

2.Easy

A radioactive tracer has a half-life of 21 days. A sample initially contains 50 mg. The decay follows dm/dt = −k·m. How much m remains after 16 days?

m₀ = 50 mgt_½ = 21 dt = 16 d

m = mg

3.Medium

Solve y″ + 7y′ + 12y = 0 with y(0) = 3 and y′(0) = 1. Find y(0.5).

y″ + 7y′ + 12y = 0y(0) = 3y′(0) = 1

y(0.5) =

4.Medium

A forged steel part at 460 °C is left to cool in a workshop at 24 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 21 min.

T₀ = 460 °CT_a = 24 °Ck = 0.03 min⁻¹t = 21 min

T = °C

5.Hard

Solve y′ + y = 4e^−0.5t with y(0) = 3. Find y(2).

y′ + y = 4e^−0.5ty(0) = 3

y(2) =

6.Hard

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

y′ + 3y/x = 2xy(1) = −3

y(2) =

Answer key

1. m = 14.14 mg
2. m = 29.49 mg
3. y(0.5) = 1.547
4. T = 256.2 °C
5. y(2) = 2.266
6. y(2) = 1.175

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