Question

Consider the partial differential equation, with the initial condition: 1 2 cup +3cºu, = 9x²y?, u(x,0) = x3 +1 Find the characteristic curves and the orthogonal trajectories and sketch both on the same graph. Find a solution of the partial differential equation with the given initial con- dition valid in the first quadrant of the (x, y)-plane. Is this solution unique? Explain.

Answer 1

solutions ooven that 24 Ux + 3x²ay - 9x² 4 2 condition u(x,0) = 23H characteristic Equation: da ay dy 3x2 dz axryz from pirst & second fraction. dy ay 3x2 3x &doc- aydy. Integrating on both sides S3x dx = faydy 24 ² G 33 3

x3-y2=0, xD a from second & Third Praction dy dz Byrdy-dz 3809 axyz بر) Integrating on both sides Saya dy = fde. 343 & C oz. Z-43 +Ca. thogonal curves x3 42-ci & 7-43-02. General solution. ca occ.). 7.43 - $ CX3-42).

2-8.46-3) UCY)y3+053-y). condition UCx,0) 20+E3). u Go). . 31 . 23 प): sacx3=y2)- Heh-ex Hence Required folution, LACY20 Cr3 -y). act).y itch-ex please Like It helps a lot for me.