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Solve a partial differential equation

Short answer

Integrate twice: u = x⁴(t+1) + x sin t + cos 2t. This satisfies the given PDE and initial conditions.

Solve the partial differential equation ∂²u/∂x² = 12x²(t + 1) subject to u(0, t) = cos 2t and uₓ(0, t) = sin t.

Integrate ∂²u/∂x² = 12x²(t + 1) twice with respect to x. The "constants" of integration are functions of t.

∂u/∂x = 4x³(t + 1) + φ(t)
u     = x⁴(t + 1) + x φ(t) + ψ(t)

Using the conditions

  • uₓ(0, t) = φ(t) = sin t
  • u(0, t) = ψ(t) = cos 2t

Answer: u(x, t) = x⁴(t + 1) + x sin t + cos 2t

Check: u_xx = 12x²(t + 1), u(0, t) = cos 2t and uₓ(0, t) = sin t.

✓ Answer checked by Claude Opus 5.5 (high effort)

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