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This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Givenequationis\frac{dy}{dx}+2xy=y\\ \Rightarrow \frac{dy}{dx}=y-2xy\Rightarrow \frac{dy}{dx}=y\left(1-2x\right)\Rightarrow \frac{dy}{y}=\left(1-2x\right)dx\\ Integratingbothsides,wehave\\ \therefore Solutionoftheequationis\\ {\displaystyle \int \frac{dy}{y}}={\displaystyle \int \left(1-2x\right)dx\Rightarrow logy=x-2.\frac{x^2}{2}+c}\\ \Rightarrow logy=x-x^2+logc\Rightarrow logy-logc=x-x^2\\ \Rightarrow log\frac{y}{c}=x-x^2\Rightarrow \frac{y}{c}=e^{x-x^2}\\ \therefore y=c.e^{x-x^2}\\ Hence,therequiredsolutionisy=c.e^{x-x^2}.\end{array}$

PW Institute of Innovation Lucknow is a private institute, established in 2023. The Institute offers various UG courses in partnership with Medhavi Skills University. Apart from the Lucknow campus, PW Institute of Innovation Lucknow has its campuses located in Pune, Bangalore, Noida, Patna, Indore and Delhi. To apply at the institute, candidates must appear for the entrance exams and fill out the application form to secure a seat.

Candidates must typically have completed Class 12 with a minimum cumulative percentage of 50 in order to meet the eligibility requirements for the UG programme. However, depending on different courses and their specialisations, this minimal requirement may change. For the MBA programme, candidates are required to get a minimum of 50% aggregate in their UG and clear the required entrance.  

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Givendifferentialequationis\left(x^2-1\right)\frac{dy}{dx}+2xy=\frac{1}{x^2-1}\\ Divindingby\left(x^2-1\right),weget\\ \frac{dy}{dx}+\frac{xy}{x^2-1}\Rightarrow \frac{1}{{\left(x^2-1\right)}^2}\\ Itisalineardifferentialequationoffirstorderandfirstdegree.\\ \therefore P=\frac{2x}{x^2-1}andQ=\frac{1}{{\left(x^2-1\right)}^2}\\ IntegratingfactorI.F=e^{{\displaystyle \int Pdx}}=e^{{\displaystyle \int \frac{2x}{x^2-1}dx}}=e^{log\left(x^2-1\right)}=\left(x^2-1\right)\\ \therefore Solutionoftheequationis\\ y\times I.F.={\displaystyle \int Q.I.F.dx+c}\\ \Rightarrow y\left(x^2-1\right)={\displaystyle \int \frac{1}{x^2-1}dx+c\Rightarrow y\left(x^2-1\right)=\frac{1}{2}log\left|\frac{x-1}{x+1}\right|}+c\\ Hence,therequiredvalueisy\left(x^2-1\right)=\frac{1}{2}log\left|\frac{x-1}{x+1}\right|+c.\end{array}$

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Givenequationis\frac{dy}{dx}=e^{-2y}\\ \Rightarrow \frac{dy}{e^{-2y}}=dx\Rightarrow e^{2y}.dy=dx\\ Integratingbothsides,weget\\ {\displaystyle \int e^{2y}.dy={\displaystyle \int dx}}\Rightarrow \frac{1}{2}{e}^{2y}=x+c\\ Puty=0andx=5\\ \Rightarrow \frac{1}{2}{e}^{2y}=5+c\Rightarrow c=\frac{1}{2}-5=-\frac{9}{2}\\ \therefore Theequationbecomes\frac{1}{2}{e}^{2y}=x-\frac{9}{2}\\ Nowputtingy=3,weget\\ \frac{1}{2}{e}^6=x-\frac{9}{2}\Rightarrow x=\frac{1}{2}{e}^6+\frac{9}{2}\\ Hence,therequiredvalueofx=\frac{1}{2}\left(e^6+9\right).\end{array}$

Yes, joining the top ranked Law colleges in Karnataka can be worthwhile for you. For that, you need to check its ROI. Listed below are some colleges along with their tuition fees and average placement package:

Disclaimer: This information is sourced from official website/ media reports/ NIRF website and may vary.

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Equationofallnonverticallinesarey=mx+c\\ Differentiatingwithrespecttox,weget\frac{dy}{dx}=m\\ Againdifferentiatingw.r.t.xwehave\frac{d^{2}y}{d{x}^2}=0\\ Hence,therequiredequationis\frac{d^{2}y}{d{x}^2}=0.\end{array}$

Hi, joining Engineering colleges accepting Andhra Pradesh EAMCET scores in Andhra Pradesh can be worthwhile. For that, you need to check the ROI which are mentioned below:

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Yes, it is estimated that joining the top ranked Law colleges in Karnataka can be expensive as most colleges have tuition fees of more than INR 5 lakh. Listed below are the top expensive Law colleges in Karnataka:

Disclaimer: This information is sourced from official website/ media reports/ NIRF website and may vary.

UCSD provides over 160 undergraduate majors to choose from to its students. The popular courses for a Bachelor's Degree include Cognitive Science, Anatomy, Computer Science, International and Global Studies, Econometrics & Quantitative Economics, General Psychology, General Biological Sciences, etc. 

Whereas, for a Master’s Degree, the list of popular courses at UC San Diego includes Computer Science, Financial Mathematics, Bioengineering & Biomedical Engineering, Other Management Sciences & Quantitative Methods, Management Science, International Relations & Affairs, Electrical & Electronics Engineering, etc.

Maharaja Ganga Singh University Rajasthan placement reports for the years, 2024 and 2023 have not been released yet. However, as per the AQAR report, the the highest package offered during MGSU Bikaner placements 2022 was INR 5.71 LPA. A total of 23 students were placed and the highest number of students were placed from MSc Computer Science. The top recruiters during the 2022 placement drive were, Govt AYUSH Wellness Center, Dunes Milk Processing Pvt. Ltd., ASKIN Softech, among others.

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Thegivendifferentialequationis\\ \frac{dy}{dx}=2^{y-x}\Rightarrow \frac{dy}{dx}=\frac{2^y}{2^x}\\ Separatingthevariables,weget\\ \frac{dy}{2^y}=\frac{dx}{2^x}\Rightarrow 2^{-y}dy=2^{-x}dx\\ Integratingbothsides,weget\\ {\displaystyle \int 2^{-y}dy}={\displaystyle \int 2^{-x}dx}\\ \frac{-2^{-y}}{log2}=\frac{-2^{-x}}{log2}+c\Rightarrow -2^{-y}=-2^{-x}+clog2\\ \Rightarrow -2^{-y}+2^{-x}=clog2\\ \Rightarrow 2^{-x}-2^{-y}=k\left[whereclog2=k\right]\end{array}$

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhatx\frac{dy}{dx}=y\left(logy-logx+1\right)\\ \Rightarrow x\frac{dy}{dx}=y\left[log\left(\frac{y}{x}\right)+1\right]\\ \Rightarrow \frac{dy}{dx}=\frac{y}{x}\left[log\left(\frac{y}{x}\right)+1\right]\\ Since,itisadifferentialequation\\ So,puty=vx\Rightarrow \frac{dy}{dx}=v+x.\frac{dv}{dx}\\ \Rightarrow v+x.\frac{dv}{dx}=\frac{vx}{x}\left[log\left(\frac{vx}{x}\right)+1\right]\\ \Rightarrow v+x.\frac{dv}{dx}=v\left[logv+1\right]\\ \Rightarrow x.\frac{dv}{dx}=v\left[logv+1\right]-v\Rightarrow x.\frac{dv}{dx}=v\left[logv+1-1\right]\\ \Rightarrow x.\frac{dv}{dx}=v.logv\Rightarrow \frac{dv}{vlogv}=\frac{dx}{x}\\ Integratingbothsides,wehave\\ {\displaystyle \int \frac{dv}{vlogv}={\displaystyle \int \frac{dx}{x}}}\\ Putlogv=tonL.H.S.\\ \Rightarrow \frac{1}{v}dv=dt\\ \therefore {\displaystyle \int \frac{dt}{t}}={\displaystyle \int \frac{dx}{x}}\\ \Rightarrow log\left|t\right|=log\left|x\right|+logc\\ \Rightarrow log\left|logv\right|=logxc\Rightarrow logv=xc\\ \Rightarrow log\left(\frac{y}{x}\right)=xc\\ Hence,therequiredsolutionislog\left(\frac{y}{x}\right)=xc.\end{array}$

As per NIRF ranking 2024, NLSIU Bangalore emerged as the top ranked Law college in Karnataka. Listed below are the colleges along with their rankings over the past three years:

Disclaimer: This information is sourced from official website of the ranking body listed and may vary.

Ganga University Bikaner has an active placement cell that provides placement assistance to the interested students who wish to seek jobs in their respective sectors. The 2024 placement reports of Odisha Open University for MSc have not been released yet. However, as per the AQAR report, the the highest package offered during MGSU Bikaner placements 2022 was INR 5.71 LPA. A total of 23 students were placed and the highest number of students were placed from MSc Computer Science. The students can check the 2024 statistics once the reports are released.

Hi, Andhra University College of Engineering, Andhra University, Visakhapatnam emerged as the top Engineering colleges in Andhra Pradesh accepting Andhra Pradesh EAMCET score along with their NIRF ranking 2025:

Hope this helps!

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

Hi, applicants who wish to enroll in a BTech programme must meet the prerequisites set forth by the universities. Applicants for the course may do so if they fulfill the eligibility requirements. The following are the standard requirements for admission to Andhra Pradesh's BTech colleges: