The simplest type of PDE

The simplest type of PDE
Some initial value problems can be solved by using antiderivatives partial.
The following examples explain how to find the solution of PDE .
Example 1: Solve the initial value problem u_x (x,y)=2ye^2x with u(0,y)=e^(-2y)
Solution :
u(x,y)=???2ye^2x dx? ? u(x,y)=ye^2x+f(y)
u(0,y)=e^(-2y) ? e^(-2y)=y+f(y)
f(y)=e^(-2y)-y
u(x,y)=ye^2x+e^(-2y)-y
Example 2: Solve the initial value problem u_t (x,t)=cos?t with u(x,0)=sin?x
u(x,t)=???cos?t dt? ? u(x,t)=sin?t+f(x)
u(x,0)=sin?x ? sin?x=0+f(x)
f(x)=sin?x
u(x,t)=sin?t+sin?x

Example 3: Solve the initial value problem u_xx (x,y)=0 ; 0?x?1
with u(0,y)=y^2 and u(1,y)=1
Solution :
u_x (x,y)=??0dx=f(y)
u(x,y)=??f(y)dx=xf(y)+h(y)
u(0,y)=y^2 ? 0×f(y)+h(y)=y^2
h(y)=y^2
u(1,y)=1 ? 1×f(y)+y^2=1
f(y)=1-y^2
So u(x,y)=x(1-y^2 )+y^2

Example 4: Solve the initial value problem u_xx (x,y)=6xy
with u(0,y)=y and u_x (1,y)=0
Solution :
u_x (x,y)=??6xy dx=3x^2 y+f(y)
u_x (1,y)=0 ? 0=3y+f(y)
f(y)=-3y
u_x (x,y)=3x^2 y-3y
u(x,y)=??(3x^2 y-3y) dx=x^3 y-3xy+h(y)
u(0,y)=y ? y=0-0+h(y)
h(y)= y
So u(x,y)=x^3 y-3xy+y
Example 5: Solve the initial value problem u_xy (x,y)=2x , u(0,y)=0 , u(x,0)=x^2
Solution :
u_y (x,y)=??2x dx=x^2+f^ (y)
u(x,y)=??(x^2+f^ (y)) dy=x^2 y+f(y)+h(x)
u(0,y)=0 ? 0=0+f(y)+h(0)
f(y)=-h(0) ?(1)
u(x,0)=x^2 ? x^2=0+f(0)+h(x)
h(x)=x^2-f(0) ?(2)
u(x,y)=x^2 y-h(0)+x^2-f(0)
u(x,y)=x^2 y+x^2-(h(0)+f(0))
Put y=0 in (1) we get f(0)=-h(0) ? f(0)+h(0)=0
So u(x,y)=x^2 y+x^2

Example 6: Solve the initial value problem u_xy (x,y)=6x^2 y
with u(1,y)=cos?y and u(x,0)=x^2
Solution: u_y (x,y)=???6x^2 y? dx=2x^3 y+f^ (y)
u(x,y)=??(2x^3 y+f^ (y)) dy=x^3 y^2+f(y)+h(x)
u(1,y)=cos?y ? cos?y=? y?^2+f(y)+h(1)
f(y)=cos?y-? y?^2-h(1) ?(1)
u(x,0)=x^2 ? x^2=f(0)+h(x)
h(x)=x^2-f(0) ?(2)

u(x,y)=x^3 y^2+cos?y-y^2-h(1)+x^2-f(0)
Put x=1 in (2) we get h(1)=1-f(0) ? f(0)+h(0)=1
So u(x,y)=x^3 y^2+cos?y-y^2+x^2-1

H.W: Solve the initial value problems
1. u_y (x,y)=tan?y with u(x,0)=ln?x
2. u_xx (x,y)=12x+4y with u_x (1,y)=y^3+6 and u(x,0)=2x^3+cos?y
3. u_xx (x,y)=y^2 e^xy with u(0,y)=cos?y and u(x,0)=2x+2
4. u_xy (x,y)=-sin??(x-y)-cos?(x+y) ?
with u(0,y)=cos?y-sin?y and u_x (x,0)=cos?x