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up vote 3 down vote favorite I need to solve this linear PDE: $3u_x - 4u_y = y^2$ The initial condition provided is: $ u (0,y)= sin(y)$ I need to use the Characteristic Method . I learned the method from this video. I have reached an answer. However, I am not sure if it is wright. My intermediate steps are: First constant: $c_1= y + frac{4}{3}x $ Second constant: $c_2= frac{y^3}{3} + 4u $ Using an arbitrary function G to make the relation between both constants, $c_2 =G(c_1) $ , we have that: $frac{y^3}{3} + 4u = G(y + frac{4}{3}x) $ With the initial condition we have: $G(y) = frac{y^3}{3} +4sin(y)$ After the definition of $G(y)$ above , I inputed the value of $c_1$ , having: $G(y + frac{4}{3}x) = frac{(y+frac{4}{3}x)^3}{3}+ 4sin(y+frac{4}{3}x) $ . Finally, solving for $u$ : $u(x,y) = frac{(y+frac{