Question

Verify that the equations

i). z = \sqrt{2x + a + }\sqrt{2y + b} and

ii). z2 + z^2 + \mu = 2(1 + \lambda ^-1)(x +\lambda y)

are both complete integrals of the PDE  z = \frac{1}{p} +\frac{1}{q}. Also show that the complete integral (ii)
is the envelope of the one parameter sub-system obtained by taking b= \frac{a}{\lambda } -\frac{\mu }{1+\lambda } in   the solution i).

 

13 Feb 2018
Answer :
Word Count : 511
To verify that the given equations are complete integrals of the partial differential equation (PDE) and to show that the second complete integral is the envelope of the one-parameter sub-system, we need to break this down step by step. ### Given PDE: \[ z = \frac{1}{p} + \frac{1}{q} \] Where \( p = \frac{\partial z}{\partial x} \) and \( q = \frac{\partial z}{\partial y} \). Now we will analyze the two given equations. ### First Equation: \[ z = \sqrt{2x + a + \sqrt{2y + b}} \] #### Step 1: Find the partial derivatives of \( z \): To check if this equation is a complete integral of the given PDE, we
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