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″ + 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 =
2.Easy
t (days)m (mg)24 dm = ?

A radioactive tracer has a half-life of 19 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 24 days?

m0 = 10 mgt½ = 19 dt = 24 d
m = mg
3.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 5 and y′(0) = -2. Find y(2).

y″ + 7y′ + 12y = 0y(0) = 5y′(0) = -2
y(2) =
4.Medium
t (min)T (°C)29Ta​282 °C361 °C

A forged steel part at 480 °C is left to cool in a workshop at 30 °C. After 29 min its temperature is 282 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 361 °C?

T0 = 480 °CTa = 30 °CT(29) = 282 °CT = 361 °C
t = min
5.Hard
tyy(0) = 2y(1.5) = ?

Solve y″ + y′ + 25.25y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + y′ + 25.25y = 0y(0) = 2y′(0) = 0
y(1.5) =
6.Hard
tyy(0) = 1y(0.75) = ?

Solve y″ + 2y′ + 17y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 2y′ + 17y = 0y(0) = 1y′(0) = 0
y(0.75) =

Answer key

  1. 1. r = −2
  2. 2. m = 4.166 mg
  3. 3. y(2) = 0.04026
  4. 4. t = 15.36 min
  5. 5. y(1.5) = 0.4161
  6. 6. y(0.75) = −0.451

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