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)8 dm = ?

A radioactive tracer has a half-life of 3 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 8 days?

m0 = 100 mgt½ = 3 dt = 8 d
m = mg
3.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 5 and y′(0) = 2. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 5y′(0) = 2
y(1.5) =
4.Medium
t (min)T (°C)46Ta​140 °C

A forged steel part at 320 °C is left to cool in a workshop at 21 °C. After 46 min its temperature is 140 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 320 °CTa = 21 °CT(46) = 140 °C
k = min⁻¹
5.Hard
tyy(0) = 2y(1.5) = ?

Solve y″ + 2y′ + 10y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 2y′ + 10y = 0y(0) = 2y′(0) = 0
y(1.5) =
6.Hard
xy12(1, 2)y(2) = ?

Solve the initial value problem y′ + 3y/x = 4x for x > 0, with y(1) = 2. Find y(2).

y′ + 3y/x = 4xy(1) = 2
y(2) =

Answer key

  1. 1. r = −2
  2. 2. m = 15.75 mg
  3. 3. y(1.5) = 3.459
  4. 4. k = 0.02003 min⁻¹
  5. 5. y(1.5) = −0.2395
  6. 6. y(2) = 3.35

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