Verify that u(x,t)=e^(−4κt)sin(2x) solves u_t=κu_xx.
Answer and explanation
u_t=−4κe^(−4κt)sin(2x) and u_xx=−4e^(−4κt)sin(2x), so u_t=κu_xx.
Credit both derivatives and the matching factor.
Subjects
Partial differential equations
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Sample exercises
On 0<x<π with zero endpoints, which positive integer n makes sin(nx) have one interior zero?
Answer and explanation
2
The n=2 mode has its single interior zero at π/2.
Which heat mode decays faster for the same positive κ: sin x or sin 3x?
- Neither decays
- sin 3x
- They decay identically
- sin x
Answer and explanation
sin 3x
The decay exponents scale with n², so 9κ exceeds κ.
What initial data are normally required for a second-order-in-time wave equation?
Answer and explanation
Initial displacement and initial velocity, together with suitable boundary conditions.
Credit two time initial conditions and domain boundaries.
For a string with L=2 and c=4, write the fundamental angular frequency as aπ and find a.
Answer and explanation
2
ω₁=cπ/L=2π.
Does the zero function solve homogeneous heat and wave equations with zero data?
- No, solutions cannot vanish
- Yes
- Only at one point
- Only if κ is negative
Answer and explanation
Yes
All derivatives and boundary/initial values are zero.
Study notes
PDE modes, boundaries and initial data
A sine mode for u_t=κu_xx decays as exp(−κk²t).
Advanced universityPDE modes, boundaries and initial data · revision notes
A sine mode for u_t=κu_xx decays as exp(−κk²t).
Advanced universityDecay of a heat-equation mode
For u_t=κu_xx on 0<x<π with zero endpoint values and initial sine mode.
Advanced universityDecay of a heat-equation mode · formula reference
For u_t=κu_xx on 0<x<π with zero endpoint values and initial sine mode.
Advanced universityAngular frequency of a wave-equation mode
For a string with fixed endpoints, length L, wave speed c, and positive integer mode n.
Advanced universityAngular frequency of a wave-equation mode · formula reference
For a string with fixed endpoints, length L, wave speed c, and positive integer mode n.
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