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)18 dm = ?

A radioactive tracer has a half-life of 22 days. A sample initially contains 40 mg. The decay follows dm/dt = −k·m. How much m remains after 18 days?

m0 = 40 mgt½ = 22 dt = 18 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
3.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 5 and y′(0) = 6. Find y(1).

y″ + 7y′ + 10y = 0y(0) = 5y′(0) = 6
y(1) =
4.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 3 and y′(0) = 0. Find y(2).

y″ + 6y′ + 9y = 0y(0) = 3y′(0) = 0
y(2) =
5.Hard
tyK = 2y(1) = ?

A system obeys y″ + 2y′ + 17y = 34 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 2y′ + 17y = 34y(0) = 0y′(0) = 0
y(1) =
6.Hard
t (min)m (kg)70 minm = ?

A tank holds 100 L of pure water. Brine with 0.25 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 70 min?

V = 100 Lm0 = 0 kgc = 0.25 kg/Lr = 2 L/min
m = kg

Answer key

  1. 1. m = 22.69 mg
  2. 2. r = −3
  3. 3. y(1) = 1.363
  4. 4. y(2) = 0.05205
  5. 5. y(1) = 2.62
  6. 6. m = 18.84 kg

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