Differential Equation – JEE Main Previous Year Question with Solutions
JEE Main Previous Year Question of Math with Solutions are available at eSaral. Practicing JEE Main Previous Year Papers Questions of mathematics will help the JEE aspirants in realizing the question pattern as well as help in analyzing weak & strong areas. eSaral helps the students in clearing and understanding each topic in a better way. eSaral is providing complete chapter-wise notes of Class 11th and 12th both for all subjects. Besides this, eSaral also offers NCERT Solutions, Previous year questions for JEE Main and Advance, Practice questions, Test Series for JEE Main, JEE Advanced and NEET, Important questions of Physics, Chemistry, Math, and Biology and many more. Download eSaral app for free study material and video tutorials.
Q. The differential equation which represents the family of curves $y=c_{1} e^{c_{2} x},$ where $c_{1}$ and $c_{2}$ are arbitrary constants, is :- (1) yy” = y’ (2) $\mathrm{yy}^{\prime \prime}=\left(\mathrm{y}^{\prime}\right)^{2}$ (3) $y^{\prime}=y^{2}$ (4) $y^{\prime \prime}=y^{\prime} y$ [AIEEE-2009]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (2)

Q. Solution of the differential equation $\cos x$ dy $=y(\sin x-y) d x, 0<x<\frac{\pi}{2}$ is : (1) sec x = (tan x + c) y (2) y sec x = tan x + c (3) y tan x = sec x + c (4) tan x = (sec x + c) y [AIEEE-2010]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (1)

Q. If $\frac{d y}{d x}=y+3>0$ and $y(0)=2,$ then $y(\ln 2)$ is equal to $:-$ (1) 13              (2) –2              (3) 7               (4) 5 [AIEEE-2011]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (3) $\frac{d y}{d x}=y+3>0 \quad y(0)=2, y(\log 2)=?$ $\int \frac{d y}{y+3}=\int d x$ $\log |y+3|=x+c$ $\mathrm{y}(0)=2$ $\log |2+3|=0+\mathrm{c} \Rightarrow \mathrm{c}=\log 5$ y. $(\log 2)=?$ $\log |y+3|=\log 2+\log 5$ $\log |y+3|=\log 10$ $y+3=10$ $y=7$

Q. Let I be the purchase value of an equipment and $V(t)$ be the value after it has been used for t years. The value $\mathrm{V}(\mathrm{t})$ depreciates at a rate given by differential equation $\frac{\mathrm{dV}(\mathrm{t})}{\mathrm{dt}}=-\mathrm{k}(\mathrm{T}-\mathrm{t})$ where $\mathrm{k}>0$ is a constant and $\mathrm{T}$ is the total life in years of the equipment. Then the scrap value $\mathrm{V}(\mathrm{T})$ of the equipment is :-. (1) $\mathrm{I}-\frac{\mathrm{k}(\mathrm{T}-\mathrm{t})^{2}}{2}$ (2) $\mathrm{e}^{-\mathrm{kT}}$ (3) $\mathrm{T}^{2}-\frac{\mathrm{I}}{\mathrm{k}}$ (4) $\mathrm{I}-\frac{\mathrm{k} \mathrm{T}^{2}}{2}$ [AIEEE-2011]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (4) $\frac{\mathrm{d} \mathrm{V}}{\mathrm{dt}}=-\mathrm{k}(\mathrm{T}-\mathrm{t})$ $\int \mathrm{d} \mathrm{V}=\int-\mathrm{k}(\mathrm{T}-\mathrm{t}) \mathrm{dt}$ $V=-k\left[T t-\frac{t^{2}}{2}\right]+C$ At $\mathrm{t}=0 \quad \mathrm{V}=\mathrm{I} \Rightarrow \mathrm{C}=\mathrm{I}$ $\mathrm{V}=-\mathrm{kt}\left(\mathrm{T}-\frac{\mathrm{t}}{2}\right)+\mathrm{I}$ $V(T)=-k T\left(T-\frac{T}{2}\right)+I=\frac{-k T^{2}}{2}+I$

Q. The curve that passes through the point (2, 3), and has the property that the segment of any tangent to it lying between the coordinate axes is bisected by the point of contact, is given by : (1) $\left(\frac{x}{2}\right)^{2}+\left(\frac{y}{3}\right)^{2}=2$ (2) 2y – 3x = 0 (3) $y=\frac{6}{x}$ (4) $x^{2}+y^{2}=13$ [AIEEE-2011]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (3) Equation of tangent at $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ is $y-y_{1}=\frac{d y_{1}}{d x_{1}}\left(x-x_{1}\right)$ $\mathrm{x}-$ intercept $=\mathrm{x}_{1}-\mathrm{y}_{1} \frac{\mathrm{d} \mathrm{x}_{1}}{\mathrm{d} y_{1}}$ According to question $x_{1}=\frac{x_{1}-y_{1} \frac{d x_{1}}{d y_{1}}}{2}$ $\Rightarrow \mathrm{x}_{1}=-\mathrm{y}_{1} \frac{\mathrm{d} \mathrm{x}_{1}}{\mathrm{d} y_{1}}$ $\int \frac{d y}{y}=\int-\frac{d x}{x}$ $\Rightarrow \quad \ell n y=-\ell n x+\ell n c$ $\Rightarrow \quad \mathrm{y}=\frac{\mathrm{c}}{\mathrm{x}} \quad \Rightarrow \quad \mathrm{xy}=\mathrm{c}$ Now at $x=2, y=3$ $\Rightarrow \quad c=6$ $\therefore \quad x y=6 \quad \Rightarrow \quad y=\frac{6}{x}$

Q. Consider the differential equation $\mathrm{y}^{2} \mathrm{d} \mathrm{x}+\left(\mathrm{x}-\frac{1}{\mathrm{y}}\right) \mathrm{d} \mathrm{y}=0 .$ It $\mathrm{y}(1)=1,$ then $\mathrm{x}$ is given by : (1) $1-\frac{1}{y}+\frac{\frac{1}{e^{y}}}{e}$ (2) $4-\frac{2}{y}-\frac{\frac{1}{e^{y}}}{e}$ (3) $3-\frac{1}{y}+\frac{\frac{1}{e^{y}}}{e}$ (4) $1+\frac{1}{y}-\frac{\frac{1}{e^{y}}}{e}$ [AIEEE-2011]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (4) $\mathrm{y}^{2} \mathrm{d} \mathrm{x}+\left(\mathrm{x}-\frac{1}{\mathrm{y}}\right) \mathrm{d} \mathrm{y}=0$ $\Rightarrow \quad \mathrm{y}^{2} \frac{\mathrm{d} \mathrm{x}}{\mathrm{d} y}+\mathrm{x}=\frac{1}{\mathrm{y}} \quad \Rightarrow \quad \frac{\mathrm{d} \mathrm{x}}{\mathrm{d} y}+\frac{\mathrm{x}}{\mathrm{y}^{2}}=\frac{1}{\mathrm{y}^{3}}$ Integrating factor (I.F.) $=\mathrm{e}^{\int \frac{1}{\mathrm{y}^{2}} \mathrm{d} \mathrm{y}}=\mathrm{e}^{-1 / \mathrm{y}}$ General solution is – \[ \mathbf{x} \cdot \mathrm{e}^{-1 / \mathrm{y}}=\int \frac{1}{y^{3}} e^{-1 / y} \mathrm{d} y+c \] Let $\mathrm{I}_{1}=\int \frac{1}{y^{3}} e^{-1 / y} \mathrm{d} y$ put $\quad \frac{-1}{y}=t$ $y^{-2} d y=d t$ $\therefore \quad \mathrm{I}_{1}=-\int \mathrm{te}^{\mathrm{t}} \mathrm{dt}$ $=-e^{t}(t-1)$ $=e^{t}(1-t)$ General solution is $\mathrm{xe}^{-1 / \mathrm{y}}=\mathrm{e}^{-1 / \mathrm{y}}\left(1+\frac{1}{\mathrm{y}}\right)+\mathrm{C}$ $\Rightarrow \quad \mathrm{x}=1+\frac{1}{y}+\mathrm{Ce}^{1 / \mathrm{y}}$ Put x = 1, y = 1 $\therefore \quad 1=1+\frac{1}{1}+\mathrm{Ce}^{1 / 1}$ $\Rightarrow \quad C=-1 / \mathrm{e}$ $\therefore \quad x=1+\frac{1}{y}-\frac{e^{1 / y}}{e}$

Q. The population p(t) at time t of a certain mouse species satisfies the differential equation $\mathrm{dp}(\mathrm{t})$ $\mathrm{dt}$ = 0.5 p(t) – 450. If p(0) = 850, then the time at which the population becomes zero is : (1) ln18 (2) 2 ln18 (3) ln9 (4) $\frac{1}{2} \ln 18$ [AIEEE-2012]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (2) $\frac{\mathrm{dp}(\mathrm{t})}{\mathrm{dt}}=\frac{1}{2} \mathrm{p}(\mathrm{t})-450$ integrate $\int \frac{\mathrm{d} \mathrm{p}}{\mathrm{p}-900}=\int \frac{1}{2} \mathrm{d} \mathrm{t}$ $\ell \mathrm{n}|(\mathrm{p}-900)|=\frac{1}{2} \mathrm{t}+\mathrm{C} \quad \ldots .(1)$ given $\mathrm{t}=0 \rightarrow \mathrm{p}=850 \quad \therefore \mathrm{C}=\ell \mathrm{n} 50$ from ( 1)

Q. At present a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers x is given by $\frac{d P}{d x}=100-12 \sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is : (1) 2500              (2) 3000               (3) 3500                  (4) 4500 [JEE (Main)-2013]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (3) $\mathrm{P}=100 \mathrm{x}-12 \mathrm{x}^{3 / 2} \cdot \frac{2}{3}+\mathrm{C}$ x = 0, P = 2000 C = 2000 $\mathrm{P}_{(\mathrm{x}=25)}=2500-1000+2000=3500$

Q. If the surface area of a sphere of radius r is increasing uniformly at the rate $8 \mathrm{cm}^{2} / \mathrm{s}$, then the rate of change of its volume is : (1) proportional to $r^{2}$ (2) constant (3) proportional to r (4) proportional to $\sqrt{\mathrm{r}}$ [JEE-Main (On line)-2013]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (3) $\mathrm{S}=4 \pi \mathrm{r}^{2}$ $\frac{\mathrm{ds}}{\mathrm{dt}}=8=8 \pi \mathrm{r} \frac{\mathrm{dr}}{\mathrm{dt}} \& \quad \mathrm{v}=\frac{4}{3} \pi \mathrm{r}^{3}$ $\Rightarrow \frac{\mathrm{dv}}{\mathrm{dt}}=4 \pi \mathrm{r}^{2} \frac{\mathrm{dr}}{\mathrm{dt}}$ $\Rightarrow \frac{\mathrm{d} \mathrm{v}}{\mathrm{dt}}=4 \mathrm{r}$

Q. Consider the differential equation $\frac{\mathrm{d} y}{\mathrm{dx}}=\frac{\mathrm{y}^{3}}{2\left(\mathrm{xy}^{2}-\mathrm{x}^{2}\right)}$ Statement 1 : The substitution z $=y^{2}$ transforms the above equation into a first order homogenous differential equation. Statement 2 : The solution of this differential equation is $y^{2} e^{-\frac{y^{2}}{x}}=C$. (1) Statement 1 is false and statement 2 is true. (2) Both statements are true. (3) Statement 1 is true and statement 2 is false. (4) Both statements are false. [JEE-Main (On line)-2013]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (2) put $z=y^{2}$ $\Rightarrow \quad \frac{\mathrm{d} z}{\mathrm{dx}}=2 \mathrm{y} \frac{\mathrm{d} \mathrm{y}}{\mathrm{d} \mathrm{x}}$ $\Rightarrow \quad$ D.E. is $\frac{\mathrm{d} z}{\mathrm{dx}}=\frac{2 \mathrm{z}^{2}}{2\left(\mathrm{xz}-\mathrm{x}^{2}\right)}$ $\Rightarrow \quad z(x d z-z d x)=x^{2} d z$ $\Rightarrow \quad \int d\left(\frac{z}{x}\right)=\int \frac{d z}{z}$ $\Rightarrow \quad \frac{\mathrm{z}}{\mathrm{x}}=\ell \mathrm{nz}+\ell \mathrm{nC}$

Q. If a curve passes through the point $\left(2, \frac{7}{2}\right)$ and has slope $\left(1-\frac{1}{\mathrm{x}^{2}}\right)$ at any point $(\mathrm{x}, \mathrm{y})$ on it, then the ordinate of the point on the curve whose abscissa is $-2$ is : $(1)-\frac{5}{2}$ ( 2)$\frac{5}{2}$ $(3)-\frac{3}{2}$ ( 4)$\frac{3}{2}$ [JEE-Main (On line)-2013]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (3) $\frac{\mathrm{d} y}{\mathrm{dx}}=1-\frac{1}{\mathrm{x}^{2}}$ $\Rightarrow \quad \mathrm{y}=\mathrm{x}+\frac{1}{\mathrm{x}}+1$ $\Rightarrow \quad \mathrm{y}=-2-\frac{1}{2}+1=-\frac{3}{2}$

Q. The equation of the curve passing through the origin and satisfying the differential equation $\left(1+x^{2}\right) \frac{d y}{d x}+2 x y=4 x^{2}$ is : $(1)\left(1+x^{2}\right) y=x^{3}$ (2) $3\left(1+x^{2}\right) y=4 x^{3}$ (3) $3\left(1+x^{2}\right) y=2 x^{3}$ (4) $\left(1+x^{2}\right) y=3 x^{3}$ [JEE-Main (On line)-2013]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (2) $\int \mathrm{d}\left(\left(1+\mathrm{x}^{2}\right) \mathrm{y}\right)=\int 4 \mathrm{x}^{2} \mathrm{d} \mathrm{x}$ $\Rightarrow \quad \mathrm{y}\left(1+\mathrm{x}^{2}\right)=\frac{4 \mathrm{x}^{3}}{3}+\mathrm{C}$ passes through $(0,0)$ $\Rightarrow \quad 3 \mathrm{y}\left(1+\mathrm{x}^{2}\right)=4 \mathrm{x}^{3}$

Q. Let the population of rabbits surviving at a time t be governed by the differential equation $\frac{\mathrm{dp}(\mathrm{t})}{\mathrm{dt}}=\frac{1}{2} \mathrm{p}(\mathrm{t})-200 .$ If $\mathrm{p}(0)=100,$ then $\mathrm{p}(\mathrm{t})$ equals : (1) $400-300 \mathrm{e}^{\mathrm{t} / 2}$ (2) $300-200 \mathrm{e}^{-t / 2}$ (3) $600-500 \mathrm{e}^{t / 2}$ (4) $400-300 \mathrm{e}^{-1 / 2}$ [JEE(Main)-2014]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (1) $\frac{\mathrm{dp}(\mathrm{t})}{\mathrm{dt}}=\frac{1}{2} \mathrm{p}(\mathrm{t})-200$ $\int_{10}^{p(t)} \frac{d p(t)}{p(t)-400}=\int_{0}^{t} \frac{d t}{2} \Rightarrow \log \left|\frac{p(t)-400}{-300}\right|=\frac{t}{2}$ $|p(t)-400|=300 e^{t / 2}$ $400-p(t)=300 e^{t / 2}$ $\therefore \mathrm{p}(\mathrm{t})=400-300 \mathrm{e}^{\mathrm{t} / 2}$

Q. Let $\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $(\mathrm{x} \log \mathrm{x}) \frac{\mathrm{d} \mathrm{y}}{\mathrm{dx}}+\mathrm{y}=2 \mathrm{x} \log \mathrm{x},(\mathrm{x} \geq 1)$ Then $\mathrm{y}(\mathrm{e})$ is equal to : (1) 2            (2) 2e            (3) e              (4) 0 [JEE(Main)-2015]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (1) $(x \log x) \frac{d y}{d x}+y=2 x \log x,(x \geq 1)$ $\Rightarrow \frac{d y}{d x}+\frac{y}{x \log x}=2$ I.F. $=e^{\int \frac{d x}{x \log x}}=e^{\log (\log x)}=\log x$ Solution is $y \log x=\int 2 \log x d x$

Q. If a curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ passes through the point $(1,-1)$ and satisfies the differential equation, $\mathrm{y}(1+\mathrm{xy}) \mathrm{dx}=\mathrm{x}$ dy, then $\mathrm{f}\left(-\frac{1}{2}\right)$ is equal to : ( 1)$\frac{4}{5}$ (2) $-\frac{2}{5}$ (3) $-\frac{4}{5}$ ( 4)$\frac{2}{5}$ [JEE(Main)-2016]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (1) Given differential equation $\mathrm{ydx}+\mathrm{xy}^{2} \mathrm{dx}=\mathrm{xdy}$ $\Rightarrow \frac{\mathrm{xdy}-\mathrm{ydx}}{\mathrm{y}^{2}}=\mathrm{xdx}$ $\Rightarrow-\mathrm{d}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)=\mathrm{d}\left(\frac{\mathrm{x}^{2}}{2}\right)$ Integrating we get $-\frac{x}{y}=\frac{x^{2}}{2}+C$ $\because$ It passes through $(1,-1)$ $\therefore \quad 1=\frac{1}{2}+\mathrm{C} \Rightarrow \mathrm{C}=\frac{1}{2}$ $\therefore \quad x^{2}+1+\frac{2 x}{y}=0 \Rightarrow y=\frac{-2 x}{x^{2}+1}$ $\therefore \mathrm{f}\left(-\frac{1}{2}\right)=\frac{4}{5}$

Q. If $(2+\sin x) \frac{d y}{d x}+(y+1) \cos x=0$ and $y(0)=1,$ then $y\left(\frac{\pi}{2}\right)$ is equal to :- (1) $\frac{4}{3}$ ( 2)$\frac{1}{3}$ (3) $-\frac{2}{3}$ $(4)-\frac{1}{3}$ [JEE(Main)-2017]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (2) $(2+\sin x) \frac{d y}{d x}+(y+1) \cos x=0$ $\frac{d}{d x}(2+\sin x)(y+1)=0$ $(2+\sin x)(y+1)=c$ $x=0, y=1 \Rightarrow c=4$ $\mathrm{y}+1=\frac{4}{2+\sin \mathrm{x}}$ $y\left(\frac{\pi}{2}\right)=\frac{4}{3}-1=\frac{1}{3}$

Q. Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\sin \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \cos \mathrm{x}=4 \mathrm{x}, \mathrm{x} \in(0$ \pi). If $\mathrm{y}\left(\frac{\pi}{2}\right)=0,$ then $\mathrm{y}\left(\frac{\pi}{6}\right)$ is equal to : (1) $\frac{-8}{9 \sqrt{3}} \pi^{2}$ (2) $-\frac{8}{9} \pi^{2}$ (3) $-\frac{4}{9} \pi^{2}$ (4) $\frac{4}{9 \sqrt{3}} \pi^{2}$ [JEE(Main)-2018]

Download eSaral App for Video Lectures, Complete Revision, Study Material and much more...

Sol. (2)

Administrator

Leave a comment

Please enter comment.
Please enter your name.


Comments
  • April 25, 2021 at 1:39 am

    very useful in april lockdown 2021 thank you very much

    4
  • February 22, 2021 at 4:27 pm

    Well…it’s very useful to those who are practicing at home!!!!

    12
  • ram
    December 3, 2020 at 5:50 pm

    should be more questions included

    4
    • August 10, 2021 at 8:00 am

      These PYQ’s are very helpful. But it should also contain recent years’ jee questions.

      1
  • December 2, 2020 at 12:29 pm

    Good questions..Please provide more of such!!

    1
  • October 29, 2020 at 11:41 pm

    Thanks A lot for your Efforts

    1
  • October 19, 2020 at 7:14 pm

    Please give sheet question in topic wise and write topic name for each every question🙋🙋🙋🙋🙋

    0
  • MK
    October 1, 2020 at 11:34 pm

    Thanks

    0
    • December 24, 2020 at 8:41 pm

      24 December 2020(fucking small)

      0
  • September 23, 2020 at 2:48 pm

    please give 2019 and 2020 questions also

    2
  • August 20, 2020 at 10:52 pm

    Thank you so much

    0
    • November 2, 2020 at 3:05 pm

      No solution available due to the error

      0
    • November 5, 2020 at 6:33 pm

      For more details, contact here: +91-6376440597, +91-9024464479 .

      0
    • November 6, 2020 at 10:14 am

      You can ask your queries on +91-6376440597 OR +91-9024464479.  

      0
  • August 20, 2020 at 9:02 pm

    Provide for us more questions which will useful for mains

    5
  • August 15, 2020 at 10:25 am

    Very nice questions has been given by you and Iam very thankful to you sir Provide us more questions from different topics sir

    0
  • July 21, 2020 at 1:20 pm

    Ya, nice questions all. Provide us more question from different topics

    0