ASSIGNED EXERCiSES4: PDE Set 2 February, 2001

PDE Problems Copson: See also Ch.4 of the lecture notes.

1. Find the types of the following differential equations, and reduce them to normal form:

i  y^{2}r − t = 0,  (1)
(ii) y^{2}r + 2ys + t = p,  (2)
(iii) y^{2}r + t = 0,  (3)
(iv) r − 2s + 3t − q − u = 0,  (4)
(v) (1 + x^{2})^{2}r − t = 0,  (5)
(vi) x^{2}r − 2xys + y^{2} + y^{2}t = 0  (6)
(vii) x^{2}r + 2xys + y^{2}t = 0.  (7)

2. Reduce the folloeing equations to normal form:

(i)  u_{xx} + 2u_{yy} + +3u_{zz} + 4u_{tt} + 2u_{xy} + 2u_{xz} + 2u_{xt} +
+ 4u_{yz} + 4u_{yt} + 6u_{xt} = 0,  (8)
(ii)  u_{xx} + u_{zz} + 2u_{xy} + 2u_{xz} + 2u_{xt} + 2u_{zt} = 0,  (9)
(iii)  u_{xx} + u_{xy} + u_{xz} + u_{xt} = 0.  (10)

3. Prove that q^{2}r − 2pqs + p^{2}t = 0 has an intemediate integral p = qf(u), where f is an arbitrary function. Hence show that y + xf(u) = g(u), where g is another arbitrary function 4. Transform the equation x^{2}u_{xx} − y^{2}u_{yy} = 0 into one with characteristic variables. Hence show that u = f(xy) + xg(\frac{y}{x}) , where f and g are arbitrary functions.

5. Find the characteristics of r − yt = 0. reduce this equation to normal form when y > 0.

6. Find the characteristics of (1 + x^{2})r − (1 + y^{2})t = 0, and reduce this equations to its normal form.

7. Prove that x^{2}r + 2xys + y^{2}t = 0 has an intermediate integral

px + qy − u + f(\frac{y}{x}) = 0

where f is an arbitrary function. Hence show that

u = f(\frac{y}{x}) + xg(\frac{y}{x})

where g is an arbitrary function.

8. Show that the equation qr + (uq − p)a = upt has an intermediate integral p + qu = f(u) , where f is an arbitarary function. Hence solve the equation.