If x equals e to the power of t sin t comma space y equals e to the power of t cos t comma space t h e n space p r o v e space t h a t space open parentheses x plus y close parentheses squared fraction numerator d squared y space over denominator d x squared end fraction equals 2 open parentheses x fraction numerator d y over denominator d x end fraction minus y close parentheses

Asked by Aswin | 5th Jan, 2016, 05:37: PM

Expert Answer:

x equals e to the power of t sin t comma space y equals e to the power of t cos t rightwards double arrow x squared plus y squared equals e to the power of 2 t end exponent open parentheses sin squared t plus cos squared t close parentheses equals e to the power of 2 t end exponent fraction numerator d x over denominator d t end fraction equals e to the power of t open parentheses sin t plus cos t close parentheses comma space fraction numerator d squared x over denominator d t squared end fraction equals e to the power of t open parentheses sin t plus cos t close parentheses plus e to the power of t open parentheses cos t minus sin t close parentheses equals 2 e to the power of t cos t fraction numerator d y over denominator d t end fraction equals e to the power of t open parentheses cos t minus sin t close parentheses comma space fraction numerator d squared y over denominator d t squared end fraction equals e to the power of t open parentheses cos t minus sin t close parentheses minus e to the power of t open parentheses sin t plus cos t close parentheses equals negative 2 e to the power of t sin t fraction numerator d y over denominator d x end fraction equals fraction numerator d y divided by d t over denominator d x divided by d t end fraction equals fraction numerator e to the power of t open parentheses c o s t minus s i n t close parentheses over denominator e to the power of t open parentheses s i n t plus c o s t close parentheses end fraction equals fraction numerator cos t minus sin t over denominator cos t plus sin t end fraction equals fraction numerator begin display style fraction numerator y minus x over denominator e to the power of t end fraction end style over denominator begin display style fraction numerator y plus x over denominator e to the power of t end fraction end style end fraction equals fraction numerator y minus x over denominator y plus x end fraction fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator begin display style fraction numerator d x over denominator d t end fraction end style begin display style fraction numerator d squared y over denominator d t squared end fraction end style minus begin display style fraction numerator d y over denominator d t end fraction end style begin display style fraction numerator d squared x over denominator d t squared end fraction end style over denominator open parentheses begin display style fraction numerator d x over denominator d t end fraction end style close parentheses cubed end fraction equals fraction numerator e to the power of t open parentheses s i n t plus c o s t close parentheses cross times open parentheses negative 2 e to the power of t s i n t close parentheses minus e to the power of t open parentheses c o s t minus s i n t close parentheses open parentheses 2 e to the power of t c o s t close parentheses over denominator open parentheses e to the power of t open parentheses s i n t plus c o s t close parentheses close parentheses cubed end fraction S i m p l i f y i n g space w e space g e t fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator negative 2 e to the power of 2 t end exponent over denominator e to the power of 3 t end exponent open parentheses cos t plus sin t close parentheses cubed end fraction equals fraction numerator negative 2 open parentheses x squared plus y squared close parentheses over denominator open parentheses x plus y close parentheses cubed end fraction L H S equals open parentheses x plus y close parentheses squared fraction numerator d squared y over denominator d x squared end fraction equals open parentheses x plus y close parentheses squared cross times fraction numerator negative 2 open parentheses x squared plus y squared close parentheses over denominator open parentheses x plus y close parentheses cubed end fraction equals fraction numerator negative 2 open parentheses x squared plus y squared close parentheses over denominator x plus y end fraction R H S equals 2 open parentheses x fraction numerator d y over denominator d x end fraction minus y close parentheses equals 2 open parentheses x cross times fraction numerator y minus x over denominator y plus x end fraction minus y close parentheses equals 2 open parentheses fraction numerator x y minus x squared minus y squared minus x y over denominator x plus y end fraction close parentheses equals negative 2 fraction numerator x squared plus y squared over denominator x plus y end fraction L H S equals R H S space left parenthesis P r o v e d right parenthesis

Answered by satyajit samal | 6th Jan, 2016, 07:47: PM