Question

Problem 3 For each of the following equations assume a product solution $u(x,y) = X(x)Y(y)$ and (where possible) carry out separation of variables to find ordinary differential equations for $X(x)$ and $Y(y)$. Be sure to distinguish clearly between $x$ and $X$, $y$ and $Y$. 1. $yu_x = u_{yy}$ 2. $u_{xx} + 4u_{yy} = 0$ 3. $u_{xx} + u_{yy} + 4u = 0$ 4. $u_{xx} + u_{yy} + xu = 0$ 5. $u_{xx} + xu_{yy} + xu = 0$ 6. $u_{xx} + xy u_{yy} = 0$ 7. $u_{xx} + (x+2)u_{yy} = 0$ 8. $u_{xx} + (x+y)u_{yy} = 0$ 9. $\frac{1}{x^2}\frac{d}{dx}(x\frac{du}{dx}) + \frac{1}{x^2}\frac{d^2u}{dy^2} = 0$

          Problem 3 For each of the following equations assume a product solution $u(x,y) = X(x)Y(y)$ and (where possible) carry out separation of variables to find ordinary differential equations for $X(x)$ and $Y(y)$. Be sure to distinguish clearly between $x$ and $X$, $y$ and $Y$.

1. $yu_x = u_{yy}$
2. $u_{xx} + 4u_{yy} = 0$
3. $u_{xx} + u_{yy} + 4u = 0$
4. $u_{xx} + u_{yy} + xu = 0$
5. $u_{xx} + xu_{yy} + xu = 0$
6. $u_{xx} + xy u_{yy} = 0$
7. $u_{xx} + (x+2)u_{yy} = 0$
8. $u_{xx} + (x+y)u_{yy} = 0$
9. $\frac{1}{x^2}\frac{d}{dx}(x\frac{du}{dx}) + \frac{1}{x^2}\frac{d^2u}{dy^2} = 0$

Problem 3 For each of the following equations assume a product solution u(x,y) = X(x)Y(y) and (where possible) carry out separation of variables to find ordinary differential equations for X(x) and Y(y). Be sure to distinguish clearly between x and X, y and Y.

1. yux = uyy
2. uxx + 4uyy = 0
3. uxx + uyy + 4u = 0
4. uxx + uyy + xu = 0
5. uxx + xuyy + xu = 0
6. uxx + xy uyy = 0
7. uxx + (x+2)uyy = 0
8. uxx + (x+y)uyy = 0
9. (1)/(x^2)(d)/(dx)(x(du)/(dx)) + (1)/(x^2)(d^2u)/(dy^2) = 0

Added by Albert L.

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