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)3 dm = ?

A radioactive tracer has a half-life of 2 days. A sample initially contains 40 mg. The decay follows dm/dt = −k·m. How much m remains after 3 days?

m0 = 40 mgt½ = 2 dt = 3 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 8y′ + 15y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 15y = 0
r =
3.Medium
t (min)T (°C)27Ta​T = ?

A can of soda at 3 °C is taken out of the fridge into a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 27 min.

T0 = 3 °CTa = 22 °Ck = 0.05 min⁻¹t = 27 min
T = °C
4.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 2 and y′(0) = 2. Find y(1).

y″ + 6y′ + 5y = 0y(0) = 2y′(0) = 2
y(1) =
5.Hard
tyK = 5y(0.75) = ?

A system obeys y″ + y′ + 16.25y = 81.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + y′ + 16.25y = 81.25y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
xy12(1, 3)y(2) = ?

Solve the initial value problem y′ + y/x = 6x2 for x > 0, with y(1) = 3. Find y(2).

y′ + y/x = 6x2y(1) = 3
y(2) =

Answer key

  1. 1. m = 14.14 mg
  2. 2. r = −3
  3. 3. T = 17.07 °C
  4. 4. y(1) = 1.097
  5. 5. y(0.75) = 8.341
  6. 6. y(2) = 12.75

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