Non existence of singular integral type 2b equations

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Non-existence of Singular Integral: Type 2(b) equations do not have singular integrals.EXERCISE 10.5.3.Find the complete integrals of the following equations:(a)q2=yp4(b)p+q=y(c)p2q3=yAns.(a)z=ax+2a2y3/23+b(b)z=a2x+(y-a)33+b(c)z=a3x+3y4/34y2+b(c) For an equation involvingp,qandz:f(p,q,z)=0.(10.5.10)Charpit’s auxiliary equations becomedxfp=dyfq=dzpfp+qfq=dp-pfz=dq-qfz·From the last two ratios, we see thatp=aq, Inserting this in (10.5.10),f(aq,q,z)=0orq=φ(z,a).(10.5.11)Usingp=aqand (10.5.11) in (10.4.4),dz=q(adx+dy)or{φ(z,a)}-1dz=adx+dywhich on integration gives the complete integral of (10.5.10):Φ(z,a)=(ax+y)+b.(10.5.12)We may employ the substitutionq=apalso, and obtain the complete integral.Non-existence of Singular Integral: Type 2(c) equations do not have singular integrals.EXERCISE 10.5.4.Find the complete integrals of the following equations:(a)z=p2+q2(b)p3=qz
Chapter 10.Partial Differential Equations of First Order61(c)z=p2+q2+1(d)z(p2+q2+1)=1(e)4(z3+1)=9z4pq(f)q2=p2z2(1-p2)(g)p2z2+q2=1(h)9(p2z+q2)=4Ans.(a)4(a2+1)z=(x+ay+b)2(b)2z=a(x+ay+b)(c)a2+1 cosh-1z=x+ay+b(d)(a2+1)(1-z2)=(x+ay+b)2(e)a2(z3+1)=(x+a2y+b)2(f)z=a2+(x+a2y+b)2(g)z2z2+a2)+a2sinh-1(z/a)=x+ay+b(h)(z+a)3/2=x+ay+bTYPE 3: Separable Equations of the formConsider an equation of the formf(p,x)=g(q,y).(10.5.13)For this, the Charpit’s auxiliary equations reduce todxfp=dy-gq=dzpfp-qgq=dp-fx=dq-gy·From these, we have an ordinary differential equation:dpdx+fxfp=0, which can be written asfpdp+fxdx=0ordf(p,x)=0.Integrating this total differential, we get the solutionf(p,x)=a.Using this in (10.5.13),f(p,x)=a,g(q,y)=a.Solving these forpandq,p=μ(x,a),q=ν(y,a).(10.5.14)Using (10.5.14) in (10.4.4),dz=μ(x,a)dx+ν(y,a)dy,which on integration gives the complete integral of (10.5.13) asz=μ(x,a)dx+ν(y,a)dy+b.(10.5.15)
6210.5.Special forms of Nonlinear First Order Partial Differential EquationsNon-existence of Singular Integral: Type 3 equations also do not have singular integrals.EXERCISE 10.5.5.Find the complete integrals of the following equations:(a)px2=qy2(b)pq+qx=y(c)p2x2-q2y2=1(d)p2-q2=x-y(e)p+q=x+y(f)(p+q)x+pq=0(g)p2=q/xy(h)px-y2q2=1Ans.(a)xyz+a(x+y)=bxy(b)z=ax-x2+y22a+b(c)z=x3a23+2a2-1y+b(d)3z=2(x+a)3/2+2(y+a)3/2+κ(e)3z=(x+a)3+(y-a)3+κ(f)2a(a+1)z=-(a+1)x2+ay2+b(g)2z=(4x+y2)a2+b(h)z=(a2+1)logx+alogy+bEXERCISE 10.5.6.Find the general and complete integrals of the following equations:(a)px-qy=y2-x2(b)p+q=sinx+siny(c)px2-2y3q=1(d)2p-3q=z(e)p+q=1Ans.(a) Auxiliary equations aredxx=dy-y=dzy2-x2. Grouping the first two ratios, we getxy=a. Choosing(x,y,1)as multipliers, the second solution isx2+y2+2z=b. Therefore, the general integral isf(xy,x2+y2+2z)=0. Also, the complete integral isx2+y2+2z=2alogxy+b.

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Bilinear Transformations, Z

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