Solve-the-d-e-d-2-y-dt-2-dy-dt-t-y-0-

Question Number 2161 by Yozzis last updated on 05/Nov/15

S o l v e t h e d . e

\frac{d^{2} y}{{d t}^{2}} \sqrt{\frac{d y}{d t}} - \frac{t}{y} = 0 .

Answered by prakash jain last updated on 07/Nov/15

Solving as a series expansion

y = \underset{i = 1}{\sum^{\infty}} a_{i} t^{i}

\frac{d y}{d t} = \underset{i = 1}{\sum^{\infty}} {i a}_{i} t^{i - 1}

\frac{d^{2} y}{{d t}^{2}} = \underset{i = 2}{\sum^{\infty}} i (i - 1) a_{i} t^{i - 2}

y^{2} {(\frac{d^{2} y}{{d t}^{2}})}^{2} (\frac{d y}{d t}) = t^{2}

{(\underset{i = 1}{\sum^{\infty}} a_{i} t^{i - 1})}^{2} {(\underset{i = 2}{\sum^{\infty}} i (i - 1) a_{i} t^{i - 2})}^{2} (\underset{i = 1}{\sum^{\infty}} {i a}_{i} t^{i - 1}) = 1

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

(t^{0}) a_{1} \cdot 2 a_{2} \cdot a_{1} = 1

m o r e w o r k t o b e d o n e .