Complex Variables and Partial Differential Equations (3130005)

Question-4a-OR

BE | Semester-3 Summer-2020 | 27-10-2020

Q4) (a)

Find the solution of $x^{2} p + y^{2} q = z^{2}$ .

Auxiliary equation:

\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R} \Rightarrow \frac{dx}{x^{2}} = \frac{dy}{y^{2}} = \frac{dz}{z^{2}}

Group-I $\frac{dx}{x^{2}} = \frac{dy}{y^{2}}$

\int \frac{dx}{x^{2}} = \int \frac{dy}{y^{2}} \Rightarrow - \frac{1}{x} = - \frac{1}{y} + c_{1} \Rightarrow \frac{1}{y} - \frac{1}{x} = c_{1}

Group-II $\frac{dy}{y^{2}} = \frac{dz}{z^{2}}$

\int \frac{dy}{y^{2}} = \int \frac{dz}{z^{2}} \Rightarrow - \frac{1}{y} = - \frac{1}{z} + c_{2} \Rightarrow \frac{1}{z} - \frac{1}{y} = c_{2}

Solution:

f (\frac{1}{y} - \frac{1}{x}, \frac{1}{z} - \frac{1}{y}) = 0

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