Complex Variables and Partial Differential Equations (3130005)

Question-5a

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

Q5) (a)

Solve $(2 D^{2} - 5 DD' + D'^{2}) z = 24 (y - x)$

(2 D^{2} - 5 DD' + 2 D'^{2}) z = 24 (y - x)

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}

So,

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

z_{p} = \frac{1}{2 D^{2} - 5 DD' + 2 D'^{2}} 24 (y - x)

\Rightarrow z_{p} = \frac{24}{2 {(- 1)}^{2} - 5 (- 1) (1) + 2 {(1)}^{2}} \iint u du du; u = y - x

\Rightarrow z_{p} = \frac{24}{9} \frac{u^{3}}{6}

\Rightarrow z_{p} = \frac{4}{9} {(y - x)}^{3}

Solution:

z = z_{c} + z_{p}

\Rightarrow z = f_{1} (y + 2 x) + f_{2} (y + \frac{x}{2}) + \frac{4}{9} {(y - x)}^{3}

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