The general 2nd order linear PDE

The general 2nd order linear PDE
The general second order linear PDE has the following form
Au_xx+Bu_xy+Cu_yy+Du_x+Eu_y+Fu=G
where the coefficients A,B,C,D,E,F and G are in general functions of the independent variables x,y, but do not depend on the unknown function u.
In this lecture we discuss only four cases as follows:
(1) When B=C=E=F=0 the PDE becomes Au_xx+Du_x=G
(2) When A=C=E=F=0 the PDE becomes Bu_xy+Du_x=G
(3) When A=C=D=F=0 the PDE becomes Bu_xy+Eu_y=G
(4) When A=B=D=F=0 the PDE becomes Cu_yy+Eu_y=G
We can solve the PDE by assuming that v=u_x in cases (1)& (2)
and v=u_y in cases (3)& (4). Then the PDE becomes linear ODE. The following examples explain how to solve the PDE.
Example 1: Solve u_xx-yu_x=2y with u(1,y)=-2 and u_x (0,y)=y-2
Solution? Put u_x=v ? u_xx=dv/dx
So dv/dx-yv=2y
Which is a linear ODE with P(x)=-y and Q(x)=2y
So,the integrating factor is I.F=e^??? P(x)dx?=e^(???-y? dx)=e^(-yx)
Then v=1/(I.F) ???I.F ? Q(x)dx ? v=1/e^(-yx) ???e^(-yx) ? 2y dx
v=e^yx (-2e^(-yx)+f(y)) ? v=-2+e^yx f(y)
u_x=v ? u_x=-2+e^yx f(y)
u_x (0,y)=y-2 ? y-2=-2+f(y) ? f(y)=y
u_x=-2+ye^yx
u(x,y)=??(-2+ye^yx )dx=-2x+e^yx+g(y)
u(1,y)=-2 ? -2=-2+e^y+g(y) ? g(y)=e^y
Then u(x,y)=-2x+e^yx+e^y
Example 2: Solve u_xy+u_x=1 with u(0,y)=0 and u_x (x,0)=sin?x
Solution?Put u_x=v ? u_xy=dv/dy ? dv/dy+v=1
Which is a linear ODE with P(y)=1 and Q(y)=1
So,the integrating factor is I.F=e^??? P(y)dy?=e^??dy=e^y
Then v=1/(I.F) ???I.F ? Q(y)dy ? v=e^(-y) ???e^y ? dy
v=e^(-y) (e^y+f^ (x)) ? v=1+e^(-y) f(x) ? u_x=1+e^(-y) f(x)
u_x (x,0)=sin?x ? sin?x=1+f(x) ? f(x)=-1+sin?x
Then u(x,y)=??(1+e^(-y) (-1+sin?x )) dx=??(1-e^(-y)+e^(-y) sin?x ) dx
u(x,y)=x-xe^(-y)-e^(-y) cos?x+g(y)
u(0,y)=0 ? g(y)=e^(-y)
u(x,y)=x-xe^(-y)-e^(-y) cos?x+e^(-y)
Example 3: Solve xu_xy+2u_y=9y^2 x with u_y (1,y)=0 and u(x,0)=1
Solution? Put u_y=v ? u_xy=dv/dx
xu_xy+2u_y=y^2? x dv/dx+2v=9y^2 x ? dv/dx+( 2 )/x v=9y^2
I.F=e^??? P(x)dx?=e^(??( 2 )/x dx)=e^(2 ln?x )=e^ln??x^2 ? =x^2
v=1/x^2 ???9x^2 ? y^2 dx=1/x^2 (3x^3 y^2+f(y)) ? u_y=3xy^2+1/x^2 f(y)
u_y (1,y)=0 ? f(y)=-3y^2
u_y=3xy^2-(3y^2)/x^2
u(x,y)=??(3xy^2-(3y^2)/x^2 ) dy=xy^3+y^3/x^2 +g(x)
u(x,0)=1 ? g(x)=1
u(x,y)=xy^3+y^3/x^2 +1
Example 4:Solve u_yy+2u_y=2 sin?x with u(x,0)=0 and u_y (x,0)=3 sin?x
Solution? Put u_y=v ? u_yy=dv/dy
dv/dy+2v=2 sin?x
I.F=e^??? P(y)dy?=e^??2dy=e^2y
v=e^(-2y) ???2e^2y ? sin?x dy
u_y=e^(-2y) (e^2y sin?x+f(x))=sin?x+e^(-2y) f(x)
u_y (x,0)=3 sin?x ? 3 sin?x=sin?x+f(x) ? f(x)=2 sin?x
u_y=sin?x+2 sin?x e^(-2y)
Then u(x,y)=??(sin?x+2 sin?x e^(-2y) ) dy
u(x,y)=y sin?x-sin?x e^(-2y)+g(x)
u(x,0)=0? g(x)=sin?x
u(x,y)=y sin?x-sin?x e^(-2y)+sin?x=sin?x (y-e^(-2y)+1)


H.W: Solve the PDE
1. u_xx+3u_x=3e^y with u(1,y)=( 2 )/( 3 ) e^(y-3) and u_x (0,y)=2e^y
2. u_xy-u_x=2 with u(0,y)=tan?y and u_x (x,0)=sec^2?x-2
3. xu_xy+u_y=0 with u_y (1,y)=cos?y and u(x,( ? )/( 2 ))=0
4. u_yy-u_y=tan^(-1)?x with u(x,0)=0 and u_y (x,0)=0