Solve-the-d-e-x-2-dy-dx-xy-x-2-y-2-1-by-letting-y-1-x-1-v-where-v-is-a-function-of-x-

Question Number 2051 by Yozzi last updated on 01/Nov/15

S o l v e t h e d . e

x^{2} \frac{d y}{d x} + x y + x^{2} y^{2} = 1

b y l e t t i n g y = \frac{1}{x} + \frac{1}{v} w h e r e

v i s a f u n c t i o n o f x .

Answered by 123456 last updated on 01/Nov/15

x^{2} \frac{d y}{d x} + x y + x^{2} y^{2} = 1

y = \frac{1}{x} + \frac{1}{v}

\frac{d y}{d x} = - \frac{1}{x^{2}} - \frac{1}{v^{2}} \cdot \frac{d v}{d x}

- x^{2} (\frac{1}{x^{2}} + \frac{1}{v^{2}} \cdot \frac{d v}{d x}) + x (\frac{1}{x} + \frac{1}{v}) + x^{2} {(\frac{1}{x} + \frac{1}{v})}^{2} = 1

- 1 - \frac{x^{2}}{v^{2}} \cdot \frac{d v}{d x} + 1 + \frac{x}{v} + {(1 + \frac{x}{v})}^{2} = 1

- \frac{x^{2}}{v^{2}} \cdot \frac{d v}{d x} + \frac{x}{v} + 1 + \frac{2 x}{v} + \frac{x^{2}}{v^{2}} = 1

\frac{x^{2}}{v^{2}} \cdot \frac{d v}{d x} = \frac{3 x}{v} + \frac{x^{2}}{v^{2}}

\frac{d v}{d x} = \frac{v^{2}}{x^{2}} (\frac{3 x}{v} + \frac{x^{2}}{v^{2}})

\frac{d v}{d x} = \frac{3 v}{x} + 1

continue

Commented by prakash jain last updated on 01/Nov/15

v = \underset{i = 1}{\sum^{n}} a_{i} x^{i}

v^{'} = \underset{i = 1}{\sum^{n}} {i a}_{i} x^{i - 1}

\frac{3 v}{x} = \underset{i = 1}{\sum^{n}} 3 a_{i} x^{i - 1}

v^{'} = \frac{3 v}{x} + 1

e q u a t i n g c o e f f i i e n t s

a_{1} = 3 a_{1} + 1 \Rightarrow a_{1} = - \frac{1}{2}

2 a_{2} = 3 a_{2} \Rightarrow a_{2} = 0

3 a_{3} = 3 a_{3} \Rightarrow a_{3} = c

4 a_{4} = 3 a_{4} \Rightarrow a_{4} = 0 a l s o a_{5}, \dots a_{n} = 0

v = {c x}^{3} - \frac{x}{2}