Question

32. The complete integral of (p+q)(z- xp- yq) = 1

Answer 1

Answer:

To get the complete integral of the given nonlinear partial differential equation, we should try to obtain a second solution Page 3 58 R.V.Waghmare and P.S.Avhale ,say, h(x,y,z,p,q,a,b) of the Charpit's equations. Then the elimination of p and q will give us the complete integral; [1].

Answer 2

Answer:

z= ax+by+1/a+b

Step-by-step explanation:

• Let (p+q)(z-xp-yq) =1 --------- (1)

• pz+qz-x$p^{2}$-xpq-y$q^{2}$ -pyq-1=0

• f(x,y,z,p,q)=pz+qz-x$p^{2}$-xpq-y$q^{2}$-pyq-1 ---------------(2)

• charpit's auxillary equation are

•    dx/fp = dy/fp =dz/pfp+qfq =dp/-(fx+pfy) =dq/-(fy+qfx) ----------(3)

• now, fx= -$p^{2}$-pq  , fy = -$q^{2}$-pq , fz=p+q,fp=z-2px-xq-yq,fq=z-xp-2yq-p

Taking last two terms in equation (3),we get & subsituting fx & fy,

•        dp/-(-$p^{2}$-pq+$p^{2}$+pq) = dq/-(-$q^{2}$-pq+pq+$q^{2}$)

•       dp/0 = dq/0

• then, p=a , q=b (integrating), where a,b are constants

• so,from (1), (a+b)(z-ax-by)=1

•     z= ax+by+1/a+b

•  This is the complete integral of(1)