Complex Variables and Partial Differential Equations (3130005)

Question-4a-OR

BE | Semester-3 Winter-2019 | 26-11-2019

Q4) (a)

Solve $\frac{\partial^{2} z}{\partial x^{2}} + 2 \frac{\partial^{2} z}{\partial x \partial y} + \frac{\partial^{2} z}{\partial y^{2}} = e^{2 x + 3 y}$ .

\frac{\partial^{2} z}{\partial x^{2}} + 2 \frac{\partial^{2} z}{\partial x \partial y} + \frac{\partial^{2} z}{\partial y^{2}} = e^{2 x + 3 y}

Auxiliary equation:

m^{2} + 2 m + 1 = 0 ⟹ (m + 1) (m + 1) = 0 ⟹ m = - 1, - 1

.

Complementary solution:

z_{c} = f_{1} (y - x) + x f_{2} (y - x)

Particular solution:

z_{p} = \frac{1}{D^{2} + 2 {DD}^{'} + D^{' 2}} e^{2 x + 3 y}

z_{p} = \frac{1}{{(2)}^{2} + 2 (2) (3) + {(3)}^{2}} e^{2 x + 3 y}

z_{p} = \frac{1}{25} e^{2 x + 3 y}

Solution:

z = z_{c} + z_{p} = f_{1} (y - x) + x f_{2} (y - x) + \frac{e^{2 x + 3 y}}{25}

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