Prove that $v(t,x) = xu_x(x,t) + 2tu_t(x,t)$ satisfies the heat equation using the fact that $u^{\lambda} = u(\lambda x, \lambda^2t)$ satisfies it

Question

Let $\lambda \in \mathbb{R}$ and denote $u_x(x,t) = \frac{d}{dx}u(x,t).$

Let $u$ satisfy the heat equation $u_t - u_{xx} = 0$ on $\mathbb{R}\times(0,\infty).$ Prove that $v(t,x) = xu_x(x,t) + 2tu_t(x,t)$ satisfies the heat equation using the fact that $u^{\lambda} = u(\lambda x, \lambda^2t)$ satisfies the heat equation.

My attempt:

I've proved that $u^\lambda$ satisfies the heat equation, and I can prove that $v$ satisfies it by just taking the proper derivatives. I'm just really not sure what I can do to use $u^{\lambda}$ in the proof, since the problem is asking that I use it, specifically, I suppose I should! Any hints would be appreciated!

score 1 · Accepted Answer · answered Jan 20 '20 at 23:26

1

Hint: $$v=\left.\frac{d u^{\lambda}}{d\lambda}\right|_{\lambda=1}$$

answered Jan 20 '20 at 23:26

Pythagoras

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Prove that $v(t,x) = xu_x(x,t) + 2tu_t(x,t)$ satisfies the heat equation using the fact that $u^{\lambda} = u(\lambda x, \lambda^2t)$ satisfies it

1 Answers1