Complex Variables and Partial Differential Equations (3130005)

Question-4c ( i )

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

Q4) (c ( i ))

Solve the p.d.e. $2 r + 5 s + 2 t = 0$

Solve the p.d.e.

2 r + 5 s + 2 t = 0

\Rightarrow 2 \frac{\partial^{2} z}{\partial x^{2}} + 5 \frac{\partial^{2} z}{\partial x \partial y} + 2 \frac{\partial^{2} z}{\partial y^{2}} = 0

\Rightarrow (2 D^{2} + 5 DD' + 2 D^{' 2}) z = 0

Auxiliary equation

2 m^{2} + 5 m + 2 = 0

\Rightarrow 2 m^{2} + 4 m + m + 2 = 0

\Rightarrow 2 m (m + 2) + 1 (m + 2) = 0

\Rightarrow (m + 2) (2 m + 1) = 0

\Rightarrow m = - 2, - \frac{1}{2}

Solution:

z = f_{1} (y - 2 x) + f_{2} (y - \frac{x}{2})

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