Question

Consider the partial differential equation for the function y(x, y) ay ay Әх ду? - ryu = 0 (i) State whether this equation is linear homogeneous, linear inhomogeneous, or non-linear. Justify your statement. (ii) Separate the variables in this equation. Find the separate equations for the variables x and y. (iii) Find the general solutions for each of the separated equations.

Answer 1

ay ²w 2x dy? - ny 4 zo. of of u Since product derivativer y is present in the given equation, the equation is ' non-linear". (6) Separation of variables Let Y (267) = X (0) Y (Y). Oy - X'w YC4). & DRY - Xody"Cy). Then ayz ax ... ay g24 - ny y=0 an 242 2 x'a yly) **) y" (y) -uy X (W) Yly) -0. X'cW) X (4) Y (Y) Y"(Y) = ax (+) Y Y Cy). X'cu) X (2) y Y (Y) YLY) Ylly) X'cu) À (say). a X (4) y constant 2 Ylly) are the =an we go Integrating, | x'cm) = ax &/xy" (y) = y equations fore for the variables x&y x Y"(y) =y (iii) X') » Y"(y) = ty Integrating, we get X (2) = sandata, » Ylly)= 1 kota, z)/ X(u) = 2x² Integrating again, we get Y (Y) = 1/2 Sydy + SaudytO2 3 Y (Y) = 1 + y² + aytaz y'ly) = 7 Sydyta, ta 2