NCERT Solutions for Exercise 9.4 Class 12 Maths Chapter 9- Differential Equations

NCERT solutions for exercise 9.4 Class 12 Maths chapter 9 gives an idea about how to solve differential equations of variable separable type. The given differential equations are converted to variable separable forms and are solved using integration. The NCERT solutions for Class 12 Maths chapter 9 exercise 9.4 presents such 23 questions. In Class 12 Maths chapter 9 exercise 9.4 certain cases are discussed where the differential equations are to be formed using the given data and then solve using the variable separable method. Two more methods were discussed in the coming exercises to solve the differential equations. Along with the Class 12th Maths chapter 6 exercise 9.4 there are a few more exercises in the NCERT book Maths Class 12 chapter listed below.

Also check -

  • Differential Equations exercise 9.1

  • Differential Equations exercise 9.2

  • Differential Equations exercise 9.3

  • Differential Equations exercise 9.5

  • Differential Equations exercise 9.6

  • Differential Equations miscellaneous exercise

Differential Equations Class 12 Chapter 9 Exercise: 9.4

Question:1 Find the general solution: \frac{dy}{dx} = \frac{1-\cos x}{1 + \cos x}

Answer:

Given,

\frac{dy}{dx} = \frac{1-\cos x}{1 + \cos x}

\\ \implies\frac{dy}{dx} = \frac{2sin^2\frac{x}{2}}{2cos^2\frac{x}{2}} = tan^2\frac{x}{2} \\ \implies dy = (sec^2\frac{x}{2} - 1)dx

\\ \implies \int dy = \int sec^2\frac{x}{2}dx - \int dx \\ \implies y = 2tan^{-1}\frac{x}{2} - x + C


Question:2 Find the general solution: \frac{dy}{dx} = \sqrt{4-y^2}\ (-2 < y < 2)

Answer:

Given, in the question

\frac{dy}{dx} = \sqrt{4-y^2}

\\ \implies \frac{dy}{\sqrt{4-y^2}} = dx \\ \implies \int \frac{dy}{\sqrt{4-y^2}} = \int dx

\\ (\int \frac{dy}{\sqrt{a^2-y^2}} = sin^{-1}\frac{y}{a})\\

The required general solution:

\\ \implies sin^{-1}\frac{y}{2} = x + C

Question:3 Find the general solution: \frac{dy}{dx} + y = 1 (y\neq 1)

Answer:

Given, in the question

\frac{dy}{dx} + y = 1

\\ \implies \frac{dy}{dx} = 1- y \\ \implies \int\frac{dy}{1-y} = \int dx

(\int\frac{dx}{x} = lnx)

\\ \implies -log(1-y) = x + C\ \ (We\ can\ write\ C= log k) \\ \implies log k(1-y) = -x \\ \implies 1- y = \frac{1}{k}e^{-x} \\

The required general equation

\implies y = 1 -\frac{1}{k}e^{-x}

Question:4 Find the general solution: \sec^2 x \tan y dx + \sec^2 y \tan x dy = 0

Answer:

Given,

\sec^2 x \tan y dx + \sec^2 y \tan x dy = 0

\\ \implies \frac{sec^2 y}{tan y}dy = -\frac{sec^2 x}{tan x}dx \\ \implies \int \frac{sec^2 y}{tan y}dy = - \int \frac{sec^2 x}{tan x}dx

Now, let tany = t and tanx = u

sec^2 y dy = dt\ and\ sec^2 x dx = du

\\ \implies \int \frac{dt}{t} = -\int \frac{du}{u} \\ \implies log t = -log u +logk \\ \implies t = \frac{1}{ku} \\ \implies tany = \frac{1}{ktanx}

Question:5 Find the general solution:

(e^x + e^{-x})dy - (e^x - e^{-x})dx = 0

Answer:

Given, in the question

(e^x + e^{-x})dy - (e^x - e^{-x})dx = 0

\\ \implies dy = \frac{(e^x - e^{-x})}{(e^x + e^{-x})}dx

Let,

\\ (e^x + e^{-x}) = t \\ \implies (e^x - e^{-x})dx = dt

\\ \implies \int dy = \int \frac{dt}{t} \\ \implies y = log t + C \\ \implies y = log(e^x + e^{-x}) + C

This is the general solution

Question:6 Find the general solution: \frac{dy}{dx} = (1+x^2)(1+y^2)

Answer:

Given, in the question

\frac{dy}{dx} = (1+x^2)(1+y^2)

\\ \implies \int \frac{dy}{(1+y^2)} = \int (1+x^2)dx

(\int \frac{dx}{(1+x^2)} =tan^{-1}x +c)

\\ \implies tan^{-1}y = x+\frac{x^3}{3} + C

Question:7 Find the general solution: y\log y dx - x dy = 0

Answer:

Given,

y\log y dx - x dy = 0

\\ \implies \frac{1}{ylog y}dy = \frac{1}{x}dx

let logy = t

=> 1/ydy = dt

\\ \implies \int \frac{dt}{t} = \int \frac{1}{x}dx \\ \implies \log t = \log x + \log k \\ \implies t = kx \\ \implies \log y = kx

This is the general solution

Question:8 Find the general solution: x^5\frac{dy}{dx} = - y^5

Answer:

Given, in the question

x^5\frac{dy}{dx} = - y^5

\\ \implies \int \frac{dy}{y^5} = - \int \frac{dx}{x^5} \\ \implies \frac{y^{-4}}{-4} = -\frac{x^{-4}}{-4} + C \\ \implies \frac{1}{y^4} + \frac{1}{x^4} = C

This is the required general equation.

Question:9 Find the general solution: \frac{dy}{dx} = \sin^{-1}x

Answer:

Given, in the question

\frac{dy}{dx} = \sin^{-1}x

\implies \int dy = \int \sin^{-1}xdx

Now,

\int (u.v)dx = u\int vdx - \int(\frac{du}{dx}.\int vdx)dx

Here, u = \sin^{-1}x and v = 1

\implies y = \sin^{-1}x .x - \int(\frac{1}{\sqrt{1-x^2}}.x)dx

\\ Let\ 1- x^2 = t \\ \implies -2xdx = dt \implies xdx = -dt/2

\\ \implies y = x\sin^{-1}x+ \int(\frac{dt}{2\sqrt{t}}) \\ \implies y = x\sin^{-1}x + \frac{1}{2}.2\sqrt{t} + C \\ \implies y = x\sin^{-1}x + \sqrt{1-x^2} + C

Question:10 Find the general solution e^x\tan y dx + (1-e^x)\sec^2 y dy = 0

Answer:

Given,

e^x\tan y dx + (1-e^x)\sec^2 y dy = 0

\\ \implies e^x\tan y dx = - (1-e^x)\sec^2 y dy \\ \implies \int \frac{\sec^2 y }{\tan y}dy = -\int \frac{e^x }{(1-e^x)}dx

\\ let\ tany = t \ and \ 1-e^x = u \\ \implies \sec^2 ydy = dt\ and \ -e^xdx = du

\\ \therefore \int \frac{dt }{t} = \int \frac{du }{u} \\ \implies \log t = \log u + \log k \\ \implies t = ku \\ \implies \tan y= k (1-e^x)

Question:11 Find a particular solution satisfying the given condition:

(x^3 + x^2 + x + 1)\frac{dy}{dx} = 2x^2 + x; \ y = 1\ \textup{when}\ x = 0

Answer:

Given, in the question

(x^3 + x^2 + x + 1)\frac{dy}{dx} = 2x^2 + x

\\ \implies \int dy = \int\frac{2x^2 + x}{(x^3 + x^2 + x + 1)}dx

(x^3 + x^2 + x + 1) = (x +1)(x^2+1)

Now,

1517900463071792

1517900463878580

1517900464704155

1517900465489721

Now comparing the coefficients

A + B = 2; B + C = 1; A + C = 0

Solving these:

1517900467056911

Putting the values of A,B,C:

1517900467842360

Therefore,

1517900468626376

1517900469359944

1517900470082760

1517900470844580

151790047163329

151790047240963

1517900473174127

1517900473936125

1517900474702185

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Now, y= 1 when x = 0

1517900477940526

c = 1

Putting the value of c, we get:

1517900478708257

Question:12 Find a particular solution satisfying the given condition:

x(x^2 -1)\frac{dy}{dx} =1;\ y = 0\ \textup{when} \ x = 2

Answer:

Given, in the question

x(x^2 -1)\frac{dy}{dx} =1

\\ \implies \int dy=\int \frac{dx}{x(x^2 -1)} \\ \implies \int dy=\int \frac{dx}{x(x -1)(x+1)}

Let,

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1517900483388423

1517900484190962

Now comparing the values of A,B,C

A + B + C = 0; B-C = 0; A = -1

Solving these:

1517900484965476

Now putting the values of A,B,C

15179004857815100

1517900486545318

1517900487390967

1517900488152880

Given, y =0 when x =2

1517900489745379

1517900490509113

1517900492095338

Therefore,

\\ \implies y = \frac{1}{2}\log[\frac{4(x-1)(x+1)}{3x^2}]

\\ \implies y = \frac{1}{2}\log[\frac{4(x^2-1)}{3x^2}]

Question:13 Find a particular solution satisfying the given condition:

\cos\left(\frac{dy}{dx} \right ) = a\ (a\in R);\ y = 1\ \textup{when}\ x = 0

Answer:

Given,

\cos\left(\frac{dy}{dx} \right ) = a

\\ \implies \frac{dy}{dx} = \cos^{-1}a \\ \implies \int dy = \int\cos^{-1}a\ dx \\ \implies y = x\cos^{-1}a + c

Now, y =1 when x =0

1 = 0 + c

Therefore, c = 1

Putting the value of c:

\implies y = x\cos^{-1}a + 1

Question:14 Find a particular solution satisfying the given condition:

\frac{dy}{dx} = y\tan x; \ y =1\ \textup{when}\ x = 0

Answer:

Given,

\frac{dy}{dx} = y\tan x

\\ \implies \int \frac{dy}{y} = \int \tan x\ dx \\ \implies \log y = \log \sec x + \log k \\ \implies y = k\sec x

Now, y=1 when x =0

1 = ksec0

\implies k = 1

Putting the vlue of k:

y = sec x

Question:15 Find the equation of a curve passing through the point (0, 0) and whose differential equation is y' = e^x\sin x .

Answer:

We first find the general solution of the given differential equation

Given,

y' = e^x\sin x

\\ \implies \int dy = \int e^x\sin xdx

\\ Let I = \int e^x\sin xdx \\ \implies I = \sin x.e^x - \int(\cos x. e^x)dx \\ \implies I = e^x\sin x - [e^x\cos x - \int(-\sin x.e^x)dx] \\ \implies 2I = e^x(\sin x - \cos x) \\ \implies I = \frac{1}{2}e^x(\sin x - \cos x)

\\ \therefore y = \frac{1}{2}e^x(\sin x - \cos x) + c

Now, Since the curve passes through (0,0)

y = 0 when x =0

\\ \therefore 0 = \frac{1}{2}e^0(\sin 0 - \cos 0) + c \\ \implies c = \frac{1}{2}

Putting the value of c, we get:

\\ \therefore y = \frac{1}{2}e^x(\sin x - \cos x) + \frac{1}{2} \\ \implies 2y -1 = e^x(\sin x - \cos x)

Question:16 For the differential equation xy\frac{dy}{dx} = (x+2)(y+2) , find the solution curve passing through the point (1, –1).

Answer:

We first find the general solution of the given differential equation

Given,

xy\frac{dy}{dx} = (x+2)(y+2)

\\ \implies \int \frac{y}{y+2}dy = \int \frac{x+2}{x}dx \\ \implies \int \frac{(y+2)-2}{y+2}dy = \int (1 + \frac{2}{x})dx \\ \implies \int (1 - \frac{2}{y+2})dy = \int (1 + \frac{2}{x})dx \\ \implies y - 2\log (y+2) = x + 2\log x + C

Now, Since the curve passes through (1,-1)

y = -1 when x = 1

\\ \therefore -1 - 2\log (-1+2) = 1 + 2\log 1 + C \\ \implies -1 -0 = 1 + 0 +C \\ \implies C = -2

Putting the value of C:

\\ y - 2\log (y+2) = x + 2\log x + -2 \\ \implies y -x + 2 = 2\log x(y+2)

Question:17 Find the equation of a curve passing through the point ( 0 ,-2) given that at any point (x,y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.

Answer:

According to the question,

y\frac{dy}{dx} =x

\\ \implies \int ydy =\int xdx \\ \implies \frac{y^2}{2} = \frac{x^2}{2} + c

Now, Since the curve passes through (0,-2).

x =0 and y = -2

\\ \implies \frac{(-2)^2}{2} = \frac{0^2}{2} + c \\ \implies c = 2

Putting the value of c, we get

\\ \frac{y^2}{2} = \frac{x^2}{2} + 2 \\ \implies y^2 = x^2 + 4

Question:18 At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).

Answer:

Slope m of line joining (x,y) and (-4,-3) is \frac{y+3}{x+4}

According to the question,

\\ \frac{dy}{dx} = 2(\frac{y+3}{x+4}) \\ \implies \int \frac{dy}{y+3} = 2\int \frac{dx}{x+4} \\ \implies \log (y+3) = 2\log (x+4) + \log k \\ \implies (y+3) = k(x+4)^2

Now, Since the curve passes through (-2,1)

x = -2 , y =1

\\ \implies (1+3) = k(-2+4)^2 \\ \implies k =1

Putting the value of k, we get

\\ \implies y+3 = (x+4)^2

Question:19 The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.

Answer:

Volume of a sphere, V = \frac{4}{3}\pi r ^3

Given, Rate of change is constant.

\\ \therefore \frac{dV}{dt} = c \\ \implies \frac{d}{dt} (\frac{4}{3}\pi r ^3) = c \\ \implies \int d(\frac{4}{3}\pi r ^3) = c\int dt \\ \implies \frac{4}{3}\pi r ^3 = ct + k

Now, at t=0, r=3 and at t=3 , r =6

Putting these value:

\frac{4}{3}\pi (3) ^3 = c(0) + k \\ \implies k = 36\pi

Also,

\frac{4}{3}\pi (6) ^3 = c(3) + 36\pi \\ \implies 3c = 252\pi \\ \implies c = 84\pi

Putting the value of c and k:

\\ \frac{4}{3}\pi r ^3 = 84\pi t + 36\pi \\ \implies r ^3 = (21 t + 9)(3) = 62t + 27 \\ \implies r = \sqrt[3]{62t + 27}

Question:20 In a bank, principal increases continuously at the rate of r % per year. Find the value of r if Rs 100 double itself in 10 years (log e 2 = 0.6931).

Answer:

Let p be the principal amount and t be the time.

According to question,

\frac{dp}{dt} = (\frac{r}{100})p

\\ \implies \int\frac{dp}{p} = \int (\frac{r}{100})dt \\ \implies \log p = \frac{r}{100}t + C

\\ \implies p = e^{\frac{rt}{100} + C}

Now, at t =0 , p = 100

and at t =10, p = 200

Putting these values,

\\ \implies 100 = e^{\frac{r(0)}{100} + C} = e^C

Also,

, \\ \implies 200 = e^{\frac{r(10)}{100} + C} = e^{\frac{r}{10}}.e^C = e^{\frac{r}{10}}.100 \\ \implies e^{\frac{r}{10}} = 2 \\ \implies \frac{r}{10} = \ln 2 = 0.6931 \\ \implies r = 6.93

So value of r = 6.93%

Question:21 In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e 0.5 = 1.648).

Answer:

Let p be the principal amount and t be the time.

According to question,

\frac{dp}{dt} = (\frac{5}{100})p

\\ \implies \int\frac{dp}{p} = \int (\frac{1}{20})dt \\ \implies \log p = \frac{1}{20}t + C

\\ \implies p = e^{\frac{t}{20} + C}

Now, at t =0 , p = 1000

Putting these values,

\\ \implies 1000 = e^{\frac{(0)}{20} + C} = e^C

Also, At t=10

, \\ \implies p = e^{\frac{(10)}{20} + C} = e^{\frac{1}{2}}.e^C = e^{\frac{1}{2}}.1000 \\ \implies p =(1.648)(1000) = 1648

After 10 years, the total amount would be Rs.1648

Question:22 In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

Answer:

Let n be the number of bacteria at any time t.

According to question,

\frac{dn}{dt} = kn\ \ (k\ is\ a\ constant)

\\ \implies \int \frac{dn}{n} = \int kdt \\ \implies \log n = kt + C

Now, at t=0, n = 100000

\\ \implies \log (100000) = k(0) + C \\ \implies C = 5

Again, at t=2, n= 110000

\\ \implies \log (110000) = k(2) + 5 \\ \implies \log 11 + 4 = 2k + 5 \\ \implies 2k = \log 11 -1 =\log \frac{11}{10} \\ \implies k = \frac{1}{2}\log \frac{11}{10}

Using these values, for n= 200000

\\ \implies \log (200000) = kt + C \\ \implies \log 2 +5 = kt + 5 \\ \implies (\frac{1}{2}\log \frac{11}{10})t = \log 2 \\ \implies t = \frac{2\log 2}{ \log \frac{11}{10}}

Question:23 The general solution of the differential equation \frac{dy}{dx} = e^{x+y} is

(A) e^x + e^{-y} = C

(B) e^{x }+ e^{y} = C

(C) e^{-x }+ e^{y} = C

(D) e^{-x }+ e^{-y} = C

Answer:

Given,

\frac{dy}{dx} = e^{x+y}

\\ \implies\frac{dy}{dx} = e^x.e^y \\ \implies\int \frac{dy}{e^y} = \int e^x.dx \\ \implies -e^{-y} = e^x + C \\ \implies e^x + e^{-y} = K\ \ \ \ (Option A)

More About NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.4

There are 23 questions in the NCERT syllabus exercise 9.4 Class 12 Maths. The first 10 questions of NCERT solutions for Class 12 Maths chapter 9 exercise 9.4 is to find the general solutions. Next 12 questions of Class 12 Maths chapter 9 exercise 9.4 is to find particular solutions using the variable separable method. And last one question of Class 12th Maths chapter 6 exercise 9.4 is of objective type and is to find the general solution.

Also Read| Differential Equations Class 12th Notes

Benefits of NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.4

  • Exercise 9.4 Class 12 Maths can be used for the preparation of the topic 9.5 for the CBSE Class 12 Board Exam.

  • Solving all the exercises given in the NCERT Book will give a good score in the exam.

Also see-

  • NCERT Exemplar Solutions Class 12 Maths Chapter 9

  • NCERT Solutions for Class 12 Maths Chapter 9

NCERT Solutions Subject Wise

  • NCERT Solutions Class 12 Chemistry

  • NCERT Solutions for Class 12 Physics

  • NCERT Solutions for Class 12 Biology

  • NCERT Solutions for Class 12 Mathematics

Subject Wise NCERT Exemplar Solutions

  • NCERT Exemplar Class 12 Maths
  • NCERT Exemplar Class 12 Physics

  • NCERT Exemplar Class 12 Chemistry

  • NCERT Exemplar Class 12 Biology