Given : f(x,y,z,p,q)=2xq2z2+2x−pz
Now we use Charpit's method, for which the following integrals are needed:
∂f/∂x=2q2z2+2
∂f/∂y=0
∂f/∂/z=4xq2z−p
∂f/∂p=−z
∂f/∂q=4xqz2
Clearly, we take the easiest two terms from Charpit's equation; we get;
dx/z=dy/(−4xqz2)
Solving this we get;
−∫4xqzdx=∫dy
⟹−2qzx2=y+a
⟹q=−(y+a)/(2x2z) , where a is a constant of integration
Putting this value of q in f(x,y,z,p,q) we get;
pz=2xz2[−(y+a)/2x2z]2+2x
⟹p=2x/z+[(y+a)2/2x3z]
Now we put these obtained values in : pdx+qdy=dz
dz=(2x/z+[(y+a)2/2x3z])dx−((y+a)/2x2z)dy
zdz=2xdx+1/4d((y+a)2/x2)
Integrating we get:
z2/2=x2+[(y+a)2/4x2]+b (Answer)
where a and b are constants of integration.
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It's very good
It is possible to solve other partial differential equations, for example, 2x(q^2z^2+1)=a, pz=a, where a is a constant.
how can we integrate (4xqz dx) taking z as constant if it is dependent on x ?
Please kindly wait for a solution of this question.
It has been days still no response