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

A radioactive tracer has a half-life of 21 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 41 days?

m₀ = 80 mgt_½ = 21 dt = 41 d

m = mg

2.Easy

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.32 h⁻¹. How many cells N are there after 24 h?

N₀ = 200k = 0.32 h⁻¹t = 24 h

N =

3.Medium

A forged steel part at 440 °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 = 0.01 min⁻¹, find its temperature T after 41 min.

T₀ = 440 °CT_a = 21 °Ck = 0.01 min⁻¹t = 41 min

T = °C

4.Medium

A cup of coffee at 80 °C is left in a room at 21 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.1 min⁻¹, find its temperature T after 19 min.

T₀ = 80 °CT_a = 21 °Ck = 0.1 min⁻¹t = 19 min

T = °C

5.Hard

Solve y′ + 0.5y = 7e^−2t with y(0) = 5. Find y(1).

y′ + 0.5y = 7e^−2ty(0) = 5

y(1) =

6.Hard

A system obeys y″ + 2y′ + 5y = 10 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value y_max of its step response.

y″ + 2y′ + 5y = 10y(0) = 0y′(0) = 0

y_max =

Answer key

1. m = 20.67 mg
2. N = 432924
3. T = 299.1 °C
4. T = 29.82 °C
5. y(1) = 5.232
6. y_max = 2.416

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