Boundary value problem PDE

Question

Solve this boundary value problem: $$ u_{xx}=0,u (0,y)=y^2,u (1,y)=1 $$ Where u is a function of x and y: $$ u=u (x,y) $$ And x varies from 0 to 1: $$ 0 <x<1 $$ And y varies from: $$ -\infty <y<\infty $$

Answer 1

If we solve this equation just by integrating u: $$ u (x,y)=c_1 (y)x+c_2 (y) $$ Now if we use boundary conditions: $$u (0,y)=y^2 \rightarrow c_2=y^2$$ $$u (1,y)=1 \rightarrow (c_1+c_2)=1 \rightarrow c_1=1-y^2$$ So we have: $$ u (x,y)=(1-y^2)x+y^2 $$

Answer 2

If we use separation of variables assuming $u(x,y)=X (x)Y (y) $ then we have: $$ X''Y=0 $$ If $Y=0$ we have trivial solution which is not desirable so: $$ X''=0 $$ It means: $$ X (x)=c_1x+c_2 $$ Now if we use boundary conditions: $$u (0,x)=X (0)Y (y) \rightarrow c_2Y (y)=y^2$$ $$u (1,x)=X (1)Y (y) \rightarrow (c_1+c_2)Y (y)=1$$ Is it correct? ?