Ask a Doubt

Request a call back

Join NOW to get access to exclusive study material for best results

CBSE Class 12-science Answered

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 | 05 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 | 06 Jan, 2016, 07:47: PM

Concept Videos

Second-order derivatives

Learn to find out the second-order derivatives of the given functions.
Second order derivatives

Learn to find out the second order derivatives of the given functions.
Second order derivatives: Exam Prep Strategies ...

Learn to find out the second order derivatives of the given functions.

Practice Test

Answered Unanswered My Questions My Answers

CBSE 12-science - Maths

x=acos^3theta and y=bsin^3theta find d^2y/dx^2.

Asked by gunturiabhishek | 18 Oct, 2020, 03:40: PM

ANSWERED BY EXPERT

CBSE 12-science - Maths

if x=a(1-cosθ ),y =b sinθ , then find dy/dx and d^2y/dx^2.

Asked by Gothai | 28 Aug, 2018, 11:33: PM

ANSWERED BY EXPERT

CBSE 12-science - Maths

If x = a cos thita + b sin thita and y = a sin thita – b cos thita, show that y²d²y/dx² – x dy/dx + y = 0 explain in great detail

Asked by haroonrashidgkp | 25 Aug, 2018, 07:15: PM

ANSWERED BY EXPERT

CBSE 12-science - Maths

If x = a(cos thita + thita sin thita) and y = a(sin thita – thita cos thita), show that d²y/dx² = 1/a (sec³thita / thita) Explain in great detail

Asked by haroonrashidgkp | 25 Aug, 2018, 07:15: PM

ANSWERED BY EXPERT

CBSE 12-science - Maths

If y = {x + (x² + 1)^1/2}^mthen show that (x² + 1)d²y/dx² + x dy/dx – m²y = 0

Asked by haroonrashidgkp | 25 Aug, 2018, 07:13: PM

ANSWERED BY EXPERT

CBSE 12-science - Maths

If y = e^{a cos-1 x} , -1 <_ x _< 1 show that (1 – x²)d²y/dx² – x dy/dx – a²y = 0 explain in great detail

Asked by haroonrashidgkp | 25 Aug, 2018, 07:05: PM

ANSWERED BY EXPERT

CBSE 12-science - Maths

Find the second order derivative of sin 3x cos 5x

Asked by haroonrashidgkp | 24 Aug, 2018, 11:12: PM

ANSWERED BY EXPERT

CBSE 12-science - Maths

Find the second order derivative of e^3x sin 4x Explain in great detail

Asked by haroonrashidgkp | 24 Aug, 2018, 11:12: PM

ANSWERED BY EXPERT

CBSE 12-science - Maths

If y = sec x – tan x , show that (cos x)d²y/dx² = y² Explain in great detail

Asked by haroonrashidgkp | 24 Aug, 2018, 11:11: PM

ANSWERED BY EXPERT

CBSE 12-science - Maths

Differentiate space straight y equals square root of fraction numerator straight x squared plus 2 over denominator straight x squared plus 3 end fraction end root

Asked by Topperlearning User | 01 Aug, 2014, 01:27: PM

ANSWERED BY EXPERT

My Class Videos Tests Plans Ask a Doubt

Get Latest Study Material for Academic year 24-25 Click here

×