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″ + 6y′ + 8y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 8y = 0
r =
2.Easy
t (days)m (mg)6 dm = ?

A radioactive tracer has a half-life of 17 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 6 days?

m0 = 200 mgt½ = 17 dt = 6 d
m = mg
3.Medium
t (min)T (°C)48Ta​50 °C

A forged steel part at 300 °C is left to cool in a workshop at 25 °C. After 48 min its temperature is 50 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 300 °CTa = 25 °CT(48) = 50 °C
k = min⁻¹
4.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 2 and y′(0) = 1. Find y(1.5).

y″ + 7y′ + 12y = 0y(0) = 2y′(0) = 1
y(1.5) =
5.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 3y′ + 11.25y = 22.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 11.25y = 22.5y(0) = 0y′(0) = 0
ymax =
6.Hard
xy12(1, 1)y(2) = ?

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

y′ + y/x = 2xy(1) = 1
y(2) =

Answer key

  1. 1. r = −2
  2. 2. m = 156.6 mg
  3. 3. k = 0.04996 min⁻¹
  4. 4. y(1.5) = 0.08263
  5. 5. ymax = 2.416
  6. 6. y(2) = 2.833

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