PDES: GENERAL AND PARTICULAR SOLUTIONS
(b)y∂u
∂x−x∂u
∂y=3x, u(1,0) = 2;(c) y^2∂u
∂x+x^2∂u
∂y=x^2 y^2 (x^3 +y^3 ), no boundary conditions.20.7 Solve
sinx∂u
∂x+cosx∂u
∂y=cosxsubject to (a)u(π/ 2 ,y)=0and(b)u(π/ 2 ,y)=y(y+1).
20.8 A functionu(x, y)satisfies
2∂u
∂x+3
∂u
∂y=10,
and takes the value 3 on the liney=4x. Evaluateu(2,4).
20.9 Ifu(x, y)satisfies
∂^2 u
∂x^2− 3
∂^2 u
∂x∂y+2
∂^2 u
∂y^2=0
andu=−x^2 and∂u/∂y=0fory=0andallx, find the value ofu(0,1).
20.10 Consider the partial differential equation
∂^2 u
∂x^2− 3
∂^2 u
∂x∂y+2
∂^2 u
∂y^2=0. (∗)
(a) Find the functionu(x, y) that satisfies (∗) and the boundary conditionu=
∂u/∂y=1wheny=0forallx. Evaluateu(0,1).
(b) In which region of thexy-plane wouldube determined if the boundary
condition wereu=∂u/∂y=1wheny=0forallx>0?20.11 In those cases in which it is possible to do so, evaluateu(2,2), whereu(x, y)is
the solution of
2 y∂u
∂x−x∂u
∂y=xy(2y^2 −x^2 )that satisfies the (separate) boundary conditions given below.(a) u(x,1) =x^2 for allx.
(b)u(x,1) =x^2 forx≥ 0.
(c) u(x,1) =x^2 for 0≤x≤ 3.
(d)u(x,0) =xforx≥ 0.
(e) u(x,0) =xfor allx.
(f) u(1,√
10) = 5.
(g) u(√
10 ,1) = 5.
20.12 Solve
6∂^2 u
∂x^2− 5
∂^2 u
∂x∂y+
∂^2 u
∂y^2=14,
subject tou=2x+1 and∂u/∂y=4− 6 x, both on the liney=0.
20.13 By changing the independent variables in the previous exercise to
ξ=x+2y and η=x+3y,
show that it must be possible to write 14(x^2 +5xy+6y^2 )intheformf 1 (x+2y)+f 2 (x+3y)−(x^2 +y^2 ),
and determine the forms off 1 (z)andf 2 (z).