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
rr² + br + c?

For the equation y″ + 10y′ + 24y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 10y′ + 24y = 0
r =
2.Easy
t (days)m (mg)18 dm = ?

A radioactive tracer has a half-life of 19 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 18 days?

m0 = 200 mgt½ = 19 dt = 18 d
m = mg
3.Medium
t (min)T (°C)6Ta​82 °C

A cup of coffee at 94 °C is left in a room at 24 °C. After 6 min its temperature is 82 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 94 °CTa = 24 °CT(6) = 82 °C
k = min⁻¹
4.Medium
t (min)T (°C)19Ta​T = ?

A can of soda at 4 °C is taken out of the fridge into a room at 29 °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 = 4 °CTa = 29 °Ck = 0.1 min⁻¹t = 19 min
T = °C
5.Hard
tyK = 2y(1) = ?

A system obeys y″ + y′ + 4.25y = 8.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + y′ + 4.25y = 8.5y(0) = 0y′(0) = 0
y(1) =
6.Hard
xy13(1, −2)y(3) = ?

Solve the initial value problem y′ + y/x = 5 for x > 0, with y(1) = −2. Find y(3).

y′ + y/x = 5y(1) = −2
y(3) =

Answer key

  1. 1. r = −4
  2. 2. m = 103.7 mg
  3. 3. k = 0.03134 min⁻¹
  4. 4. T = 25.26 °C
  5. 5. y(1) = 2.229
  6. 6. y(3) = 6

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