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Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q. 1. If (a + bx) ey/x = x, then prove that Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE             ...(1)

Diff. w.r. to x, we get

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                 ...(2)

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Differentiating (3) w. r. to  x, we get

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE    ..(4)

Comparing (3) and (4) we get

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 


Q. 2. A normal is drawn at a point P(x, y) of a curve. It meets the x–axis at Q. If PQ is of constant length k, then show that the differential equation describing such curves is Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Find the equation of such a curve passing through (0, k).

Solution. The length of normla PQ to any  curve y = f (x) is given by

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

According to question

length of PQ = k

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
which is the required differential equation of given curve. On solving this D.E. we get the eqn of curve as follows

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

As it passes through (0, k) we get C = 0

∴ Eqn of curve is

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 3. Let y = f (x) be a curve passing through (1, 1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Form the differential equation and determine all such possible curves.

Ans. x +y = 2 and xy = 1, x,y > 0

Solution. Equation of the tangent to the curve y = f (x) at point

(x,y) is Y - y = f '(x)(X-x) ...(1)

The line (1) meets X-axis at  Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  and Y-axis in

Q (0, y - xf '(x))

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Area of triangle OPQ is

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

We are given that area of ΔOPQ = 2

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ (y - px)2 + 4p = 0         ...(2)
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Since OQ > 0, y - xf '(x) > 0. Also note that

p = f '(x)<0

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE       ...(3)

Differentiating (3) with respect to x, we get

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then p = c where c < 0 [∵ p  < 0]

Putting this value in (3) we get

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   ...(4)

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Putting the value of c in (4), we get

y = -x + 2, or x +y= 2

Next, putting x = (- p)-1/ 2 or - p = x-2 in (3) we get

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ x y = 1(x > 0,y > 0)

Thus, the two required curves are x +y= 2 and xy = 1, ( x > 0,y> 0).

Q. 4. Determine the equation of the curve passing through the origin, in the form y = f (x), which satisfies the differential equation Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Put 10 x +6 y = v

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Divide numerator and denomenator by  Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and put  Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
At origin x = 0,y = 0

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence, from above

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
or 4t = (5 + 3t) tan 4x  or t (4 - 3 tan 4 x) = 5 tan 4x

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 5. Let u(x) and v(x) satisfy the differential equation  Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE where p(x) f(x) and g(x) are continuous functions. If u(x1) > v(x1) for some x1 and f(x) > g(x) for all x > x1, prove that any point (x, y) where x > x1, does not satisfy the equatons y = u(x) and y = v(x).

Solution. (i) y = u (x) and y = v(x) are solutions of given differential equations.

(ii) u (x1) > v(x1) for some x1

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

From above since Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and exponential function is always +ive, then R.H.S. is +ive.

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence the function  Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is an increasing function.

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(F being increasing function)

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Hence there is no point (x,y) such that x > x1 which can satisfy the equations.

y = u (x) and y = v(x).


Q. 6. A curve passing through the point (1, 1) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x–axis. Determine the equation of the curve.

Ans. x2 + y2 – 2 x = 0,x - 1 = 0

Solution. Equation of normal is  Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then x = c, when x = 1, y = 1,c= 1. 

∴  x = 1   ...(1)

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Solution is x2 + y2 - 2x = 0            ...(2)

Hence the solutions are,

x2 + y2 - 2x = 0,x -1 = 0.


Q. 7. A country has a food deficit of 10%. Its population grows continously at a rate of 3% per year. Its annual food production every year is 4% more than that of the last year.
 Assuming that the average food requirement per person remains constant, prove that the country will become self– sufficient in food after n years, where n is the smallest integer bigger than or equal to 
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Let X0 be initial population of the country and Y0 be its initial food production. Let the average consumption be a units. Therefore, food required initially a X0. It is given

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE             ...(i)

Let X be  the population of the country in year t.

Then dX/dt = rate of change of population

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Let Y  be the food production in year t.

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Food consumption in the year t is a X 0e0.03t

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Taking log on both sides,

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Thus, the least integral value of the year n, when the country becomes self-sufficient, is the smallest integer greater than  or equal to   Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  


Q. 8. A hemispherical tank of radius 2 metres is initially full of water and has an outlet of 12 cm2 cross–sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law v(t) = 0.6 Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere v(t) and h(t) are respectively the velocity of the flow through the outlet and the height of water level above the outlet at time t, and g is the acceleration due to gravity. Find the time it takes to empty the tank. (Hint : Form a differential equation by relating the decrease of water level to the outflow).

Ans. Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Let the water level be at a height h after time t, and water level falls by dh in time dt and the corresponding volume of water gone out be dV.

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE      (∵ dh is very small)
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE(∵ as t increases, h decreases)

Now, velocity of water, Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Rate of flow of water = Av (A= 12 cm 2)

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Also from figure,

R2 = (R - h)2+r2 ⇒ r2 = 2hR-h2

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Integrating,

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 9. A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant = k > 0).
 Find the time after which the cone is empty.

Ans. H/k

Solution. Let at time t, r and h be the radius and height of cone  of water.

∴ At time t, surface area of liquid in contact with air  = πr2

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

[∵ ‘–’ve sign shows that V decreases with time.]

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

But from figure  Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  [Using similarity of Δ’s ] 

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

But at t = 0, r = R ⇒ R = 0 + C ⇒ C = R

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Now let the time at which cone is empty be T then at T, r = 0 (no liquid is left)

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 10. A curve ‘C’ passes thr ough (2, 0) an d the slope at (x, y) as Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEFind the equation of the curve. Find the area bounded by curve and x–axis in fourth quadrant.

Ans. Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. According to question

 slope of curve C at  Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
y = x( x + 1) + 3 + C (x+ 1)   ...(1)

As the curve passes through (2, 0)

∴ 0 =  2.3 +3 + C.3

⇒ C =-3
∴  Eqn. (1) becomes

y = x(x +1) + 3 - 3x-3

y = x2- 2x ...(2)

which is the required eqn of curve.
This can be written as (x - 1)2 = (y+ 1)

[Upward parabola with vertex at (1, –1), meeting x-axis at (0, 0) and (2, 0)]

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Area bounded by curve and x-axis in fourth quadrant is as shaded region in fig. given by

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 11. If length of tangen t at any point on the curve y = f(x) intecepted between the point and the x–axis is of length 1. Find the equation of the curve.

Ans. Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. We know that length  of tangent to curve y = f (x) is given by

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Differential Equations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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