Answer to Question #310692 in Differential Equations for Ronan

Question #310692

find the complete integral and singular integral of 4(1+x3)=9z4pq.


1
Expert's answer
2022-03-19T02:41:41-0400

4(1+x3)=9z2pq

f(x,y,z,p, q)=4x3- 9z2pq+4=0.............(1)


dpdFdx+PdFdz=dqdFdy+qdFdz=dzPdFdpqdFdq=dxdFdx=dydFdq\frac{dp}{\frac{dF}{dx}+P\frac{dF}{dz}}=\frac{dq}{\frac{dF}{dy}+q\frac{dF}{dz}}=\frac{dz}{-P\frac{dF}{dp}-q\frac{dF}{dq}} =\frac{dx}{-\frac{dF}{dx}} =\frac{dy}{-\frac{dF}{dq}}


dp12x218zp2q=dq18zpq2=dz9z2pq9z2pq=dx9z2q=dy9z2p\frac{dp}{12x^2 -18zp^2q}=\frac{dq}{-18zpq^2}=\frac{dz}{9z^2pq-9z^2pq} =\frac{dx}{-9z^2q}=\frac{dy}{-9z^2p}

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Solving this we get dz=0

Integrating (1), we equate q=C1

4x39z2pC1+4=04x^3-9z^2pC1+4=0


P=4x3+49z2C1P=\frac{4x^3+4}{9z^2C1}


dz=pdx+qdydz= pdx+qdy


dz=4(x3+4)9z2C1dx+C1dydz= \frac{4(x^3+4)}{9z^2C1}dx+C1dy


dzC1dy=4(x3+1)9z2C1dxdz-C1dy= \frac{4(x^3+1)}{9z^2C1}dx


d(zC1y)=4(x3+1)9z2C1d(z-C1y)=\frac{4(x^3+1)}{9z^2C1}

Integrating this we get,

log(zC1y)=log9z2C1+logC2log(z-C1y)=log9z^2C1+log C2

C2 is an arbitrary constant

ZC1y=49z2C1+C2.............(2)Z-C1y =\frac{4}{9z^2C1}+C2.............(2)

(2) is our complete integral

To get the singular integration we add 1 and 2

Differentiate 2 with respect to c1 and c2 respectively we get,


y=9z2C2y=9z^2C2

0=9z2c1,c1=0C2=y9z20=9z^2c1, c1=0 C2=\frac{y}{9z^2}

z=0

Therefore y=c2


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