Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Explain solution RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 52 maths

Explain solution RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 52 maths

Answers (1)

best_answer

Answer:

x^{2} \frac{d^{2} y}{d x^{2}}+x(1-2 n) \frac{d y}{d x}+\left(1+n^{2}\right) y=0

Hint:

You must know the derivative of

        cos(log\: x)\: and\: sin(log\: x)

Given:

        \text { If } y=x^{n}\{\operatorname{acos}(\log x)+b \sin (\log x)\} \text { prove that } x^{2} \frac{d^{2} y}{d x^{2}}+x(1-2 n) \frac{d y}{d x}+\left(1+n^{2}\right) y=0

Solution:

        \begin{aligned} &\text { Let } y=x^{n}\{a \cos (\log x)+b \sin (\log x)\}\\ &\text { Use multiplicative rule }\\ &\mathrm{UV}=\mathrm{U}^{\prime} \mathrm{V}+\mathrm{UV}^{\prime}\\ &U=x^{n}\\ &V=a \cos (\log x)+b \sin (\log x) \end{aligned}

        \begin{aligned} &\frac{d y}{d x}=n x^{n-1}\{\operatorname{acos}(\log x)+b \sin (\log x)\}+x^{n}\left\{\frac{a \sin (\log x)}{x}+\frac{b \cos (\log x)}{x}\right\} \\ &\frac{d y}{d x}=n \frac{y}{x}+\frac{x^{n}}{x}(-a \sin (\log x)+b \cos (\log x)) \\ &x \frac{d y}{d x}=n y+x^{n}(-a \sin (\log x)+b \cos (\log x)) \end{aligned}

        \begin{aligned} &\frac{d y}{d x}+x \frac{d^{2} y}{d x^{2}}=n \frac{d y}{d x}+n x^{n-1}(-a \sin (\log x)+b \cos (\log x))+x^{n}\left\{\frac{a \sin (\log x)}{x}+\frac{b \cos (\log x)}{x}\right\} \\ &\frac{d y}{d x}(1-x)+x \frac{d^{2} y}{d x^{2}}=n x^{n-1}(-a \sin (\log x)+b \cos (\log x))+x^{n}\left\{\frac{a \sin (\log x)}{x}+\frac{b \cos (\log x)}{x}\right\} \end{aligned}

        \begin{aligned} &\frac{d y}{d x}(1-n)+x \frac{d^{2} y}{d x^{2}}=n x^{n-1}(-a \sin (\log x)+b \cos (\log x))-\frac{1}{x} y \\ &\frac{d y}{d x}(1-n)+x \frac{d^{2} y}{d x^{2}}=\frac{n}{x}\left(x \frac{d y}{d x}-n y\right)-\frac{y}{x} \end{aligned}

        \begin{aligned} &\frac{d y}{d x}+x(1-n) \frac{d y}{d x}=n x \frac{d y}{d x}-n^{2} y-y \\ &\frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}(x-n x-n x)-\left(1+n^{2}\right) y=0 \\ &x^{2} \frac{d^{2} y}{d x^{2}}+x(1-2 n) \frac{d y}{d x}+\left(1+n^{2}\right) y=0 \end{aligned}

Posted by

infoexpert25

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Class 12 Business Studies Exam Analysis: Easy, balanced paper; questions from NCERT
anam-khan

CBSE Class 12 business studies, business administration exams today; Guidelines
anam-khan

CBSE 2023 Exam: Class 12 Informatics Practices, Computer Science exam analysis, students' reaction here

Latest Question