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′ + 4y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 4y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 6y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6y′ + 36y = 0
ζ =
3.Medium
tyy(0) = 3y(1) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = 0. Find y(1).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = 0
y(1) =
4.Medium
t (min)T (°C)49Ta​T = ?

A forged steel part at 360 °C is left to cool in a workshop at 28 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.01 min⁻¹, find its temperature T after 49 min.

T0 = 360 °CTa = 28 °Ck = 0.01 min⁻¹t = 49 min
T = °C
5.Hard
tyK = 1y(1) = ?

A system obeys y″ + 3y′ + 6.25y = 6.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 3y′ + 6.25y = 6.25y(0) = 0y′(0) = 0
y(1) =
6.Hard
ty2y(2) = ?

Solve y′ + 2y = 10e−t with y(0) = 5. Find y(2).

y′ + 2y = 10e−ty(0) = 5
y(2) =

Answer key

  1. 1. r = −1
  2. 2. ζ = 0.5
  3. 3. y(1) = 1.218
  4. 4. T = 231.4 °C
  5. 5. y(1) = 0.9407
  6. 6. y(2) = 1.262

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