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I'm having a bit of trouble getting my head around some notation in a question. I'm told that $u$ satisfies the heat equation $u_t - u_{xx} = 0$ and I'm asked to find the equation satisfied by $v(y, t) = u(kx, t)$ where $k$ is a constant.

So if $y = kx$ then $\partial y = k \partial x$, so that $$v_y(y, t) = (1/k)u_x(kx, t) = (1/k)u_{(kx)}(kx, t)k = u_{(kx)}(kx, t)$$ But then $v(y, t)$ just satisfies the same equation, namely $v_t - v_{yy} = 0$. Is there a mistake with the question's notation or am I doing something wrong? It seems to me that $v$ should be defined as $v(x, t) = u(kx, t) = u(y, t)$ or something like that. Then we would have $v_x(x, t) = ku_y(y, t)$, and so the equation for $v$ would be $v_t - (1/k^2) v_{xx} = 0$

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I totally agree with you. I think that the question is actually supposed to be what you think it should be: $v(x,t)=u(kx,t)$. In this case we indeed get (with $y=kx$) $$v_{xx}=k^2u_{yy}$$ and consequently $$v_t - \frac{1}{k^2}v_{xx}=0$$ With the original formulation of the question, $v$ and $u$ satisfy the same differential equation, as you have correctly pointed out.

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