Complex Variables and Partial Differential Equations (3130005)

Question-5a-OR

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

Q5) (a)

Solve $p^{3} + q^{3} = x + y$ .

∎ p^{3} + q^{3} = x + y

Take,

p^{3} - x = y - q^{3} = a

\Rightarrow p^{3} - x = a & y - q^{3} = a

\Rightarrow p^{3} = x + a & q^{3} = y - a

\Rightarrow p = {(x + a)}^{\frac{1}{3}} & q = {(y - a)}^{\frac{1}{3}}

Now,

dz = p dx + q dy

\Rightarrow \int dz = \int p dx + \int q dy

\Rightarrow \int dz = \int {(x + a)}^{\frac{1}{3}} dx + \int {(y - a)}^{\frac{1}{3}} dy

\Rightarrow z = \frac{3}{4} {(x + a)}^{\frac{4}{3}} + \frac{3}{4} {(y - a)}^{^{\frac{4}{3}}} + b

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