diff --git a/src/8_differential_equations_2.md b/src/8_differential_equations_2.md index 4afb3faa5fdeedd90f1cc6c4f14673493d4e2db3..aa07d6bad6a30bbbd095b65afc010bdb24edc9f3 100644 --- a/src/8_differential_equations_2.md +++ b/src/8_differential_equations_2.md @@ -574,11 +574,11 @@ necessary to work with numerical methods of solution. $\frac{d^2}{dx^2}$ is Hermitian - you can thus directly assume that the solutions form an orthogonal basis). -5. [:smirk:] Take the partial differential equation +5. [:smirk:] Consider the following partial differential equations, and try to make a separation ansatz $h(x,y)=f(x)g(y)$. What do you observe in each case? (Only attempt the separation, do not solve the problem fully) - $$\frac{\partial h(x,y)}{\partial x} + x \frac{\partial h(x,y)}{\partial y} = 0. $$ + (a) $$\frac{\partial h(x,y)}{\partial x} + x \frac{\partial h(x,y)}{\partial y} = 0. $$ - Try to make a separation ansatz $h(x,y)=f(x)g(y)$. What do you observe? + (b) $$\frac{\partial h(x,y)}{\partial x} + \frac{\partial h(x,y)}{\partial y} + xy\,h(x,y) = 0$$ 6. [:sweat:] We consider the Hilbert space of functions $f(x)$ defined for $x \ \epsilon \ [0,L]$ with $f(0)=f(L)=0$.