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)11 dm = ?

A radioactive tracer has a half-life of 18 days. A sample initially contains 50 mg. The decay follows dm/dt = −k·m. How much m remains after 11 days?

m0 = 50 mgt½ = 18 dt = 11 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 5y′ + 6y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 6y = 0
r =
3.Medium
t (min)T (°C)18Ta​41 °C

A cup of coffee at 83 °C is left in a room at 19 °C. After 18 min its temperature is 41 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 83 °CTa = 19 °CT(18) = 41 °C
k = min⁻¹
4.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 3 and y′(0) = 0. Find y(0.5).

y″ + 7y′ + 10y = 0y(0) = 3y′(0) = 0
y(0.5) =
5.Hard
xy12(1, −3)y(2) = ?

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

y′ + 2y/x = 2xy(1) = −3
y(2) =
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + y′ + 16.25y = 16.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 16.25y = 16.25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. m = 32.73 mg
  2. 2. r = −2
  3. 3. k = 0.05932 min⁻¹
  4. 4. y(0.5) = 1.675
  5. 5. y(2) = 1.125
  6. 6. ymax = 1.675

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