2. Given the nonlinear equation: F(x, y,u, u, Uy)= ru, - (uy)? - 2u = 0 with the Initialcondition: r(0, s) = 2o(s) = 8, y(0, s) = %(s) = s, u(0, s) = uo(s) = 0%3D%3Da) Write the Charpit equations for F(r, y, u, p, q) = 0, p = uz, = lydedyduF, (1),F, (2),dt= pF, + qF, (3),dtdtdpdq-F, - pF.dt(4),-F, - qF. (5),dtb) Write all Initial conditions for a, y, u and p, q. (Do not use, ignore, the conditionp(0, s) = po(s) = 0, q(0, s) = g0(s) = 0 which lead to the trivial solution u = 0. Use the%3D%3D%3Dother one.)c) Respectively find a(t, s), p(t, s), q(t, s), y(t, s), u(t, s) (in the given order) using theInitial conditions.d) Write the solution u interms of r and y: u(a, y).

Given the non-linear equation, Fx,y,u,ux,uy=xux-uy2-2u=0 =px-q2-2u=0…

2. Given the nonlinear equation: F(x,y, u, u, Uy) = ru, - (u,)2 - 2u = 0 with the Initial condition: r(0, s) = 1(s) = 8, y(0, s) = (s) = s, u(0, s) = uo(s) = 0 a) Write the Charpit equations for F(r, y, u,p, q) 0, p u, q= ly %3D dr dy F, (2), du F, (1), = pF, +qF, (3), dt dt dt dp dq -F, - pF. (4), = -F, - qF. (5), dt dt b) Write all Initial conditions for r, y, u and p, q. (Do not use, ignore, the condition p(0, s) = po(s) = 0, q(0, s) = q0(s) = 0 which lead to the trivial solution u = 0. Use the %3D other one.)

2. Given the nonlinear equation: F(x,y, u, u, Uy) = ru, - (u,)2 - 2u = 0 with the Initial condition: r(0, s) = 1(s) = 8, y(0, s) = (s) = s, u(0, s) = uo(s) = 0 a) Write the Charpit equations for F(r, y, u,p, q) 0, p u, q= ly %3D dr dy F, (2), du F, (1), = pF, +qF, (3), dt dt dt dp dq -F, - pF. (4), = -F, - qF. (5), dt dt b) Write all Initial conditions for r, y, u and p, q. (Do not use, ignore, the condition p(0, s) = po(s) = 0, q(0, s) = q0(s) = 0 which lead to the trivial solution u = 0. Use the %3D other one.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

2. Given the nonlinear equation: F(x, y,u, u, Uy)
= ru, - (uy)? - 2u = 0 with the Initial
condition: r(0, s) = 2o(s) = 8, y(0, s) = %(s) = s, u(0, s) = uo(s) = 0
%3D
%3D
a) Write the Charpit equations for F(r, y, u, p, q) = 0, p = uz, = ly
de
dy
du
F, (1),
F, (2),
dt
= pF, + qF, (3),
dt
dt
dp
dq
-F, - pF.
dt
(4),
-F, - qF. (5),
dt
b) Write all Initial conditions for a, y, u and p, q. (Do not use, ignore, the condition
p(0, s) = po(s) = 0, q(0, s) = g0(s) = 0 which lead to the trivial solution u = 0. Use the
%3D
%3D
%3D
other one.)
c) Respectively find a(t, s), p(t, s), q(t, s), y(t, s), u(t, s) (in the given order) using the
Initial conditions.
d) Write the solution u interms of r and y: u(a, y).

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The equations xy² + 6xzcos(u) + ye" = 12 and x³yz + xe" 24 are %3D solved for u and v as functions of x, y and z near the point P where (x,y,z)=(1,1,1) and (u, v) = (5,0). Find () 2,y at P. u O -80,00 O 12,00 O -0,17 O 64,00 O - 30,00
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