Plus Two Maths Chapter Wise Questions and Answers Chapter 9 Differential Equations are part of Plus Two Maths Chapter Wise Questions and Answers. Here we have given Plus Two Maths Chapter Wise Questions and Answers Chapter 9 Differential Equations.

Board | SCERT, Kerala |

Text Book | NCERT Based |

Class | Plus Two |

Subject | Maths Chapter Wise Questions |

Chapter | Chapter 9 |

Chapter Name | Differential Equations |

Number of Questions Solved | 50 |

Category | Kerala Plus Two |

## Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 9 Differential Equations

**Short Answer Type Questions (Score 3)**

Question 1.

Consider

i. Express as a function of y/x.1

ii. Solve the equation using the substitution y = vx. **(2) **Answer:

Question 2.

a. Find the solution of y = e^{x} + 1. **(2)**

b. Obtain the equation of the family of straight lines parallel to the line y = 2x. ** (1)**

Answer:

Question 3.

i. Write down the order and degree of

**(1) **ii. What is the integrating factor of the equation?

**(1)**

iii. Obtain its general solution. 1

Answer:

i. Order = 1, Degree = 1

Question 4.

Solve the initial value problem: **(3) **

Answer:

Question 5.

a. Find the degree of the differential equation in

**(1)**

b. Find the integrating factor of the differential equation cosx (dy/dx) + y sin x = 1. **(2) **Answer:

Question 6.

a. Find the number of arbitary constants in the general solution of a differential equation of fourth order.

b. What is the degree of the following differential equation ? ** (3) **

Answer:

a. 4

b. Degree of differencial equation

Question 7.

Solve

**(3)**

Answer:

Question 8.

solve

**(3)**

Answer:

Given

On separating the variables and integrating ,we get

Question 9.

Solve e^{x} tany dx + (1 -e^{x}) sec^{2}ydy = 0. 3 **(3)**

Answer:

Question 10.

Solve

**(3)**

Answer:

**Long Answer Type Questions**

**(Score 4)**

Question 1.

Consider the differential equation 3e^{x }tan y dx – (l+e^{x}) sec^{2} y dy = 0

i. Order of the differential equation is…. ** (1) **ii. Express the differential equation in variable separable form.

**(1)**

iii. Solve the given equation.

**(2)**

Answer:

i. 1

Question 2.

Solve

i. **(2)**

ii. **(2)**

Answer:

Question 3.

Given y dx – x dy + (log x) dx = 0

i. Express the given equation in the form

**(1)**

ii. Find the integrating factor. **(1)**

iii. Solve the given differential equation. **(2) **Answer:

i. y dx – x dy + log x dx

dividing by xdx

Question 4.

Consider the differential equation

i. Find its integrating factor. **(2)**

ii. Solve the differential equation. **(2) **Answer:

Question 5.

a. Find the solution of e^{x }cosy dx – e^{x }siny dy=0. **(3)**

b. Find the order of the differential equation

**(1) **Answer:

a. e

^{x}cos y = c

e

^{x}cos y dx = e

^{x}sin y dy

∫ tan y dy =∫ dx

x = log sec y + log c. e

^{x}= sec y. c

⇒ e

^{x}cos y = c.

b. order of highest order derivative is 2.

Question 6.

Show that the function _{ }solution of the differential equation

**(4) **Answer:

Question 7.

Form the differential equation corresponding to y^{2} = a(b-x)(b+x) by eliminating a and b. **(4)**

Answer:

Question 8.

Find the equation of the curve passing through the point (-2,3) given that the slope of the tangent to the curve at point (x,y) is **(4) **Answer:

Question 9.

Find the differential equation of all the circles touching the x-axis at origin. **(4) **Answer:

Let radius = a, centre is (0,a)

Equation is (x-0)

^{2}+ (y – a )

^{2}= a

^{2}

Question 10.

Form the differential equation of the family of circles having centre on the y-axis and radius 3 units. **(4) **Answer:

Centre is (0,a) and radius 3.

∴ Equation of circle is (x-0)

^{2}+ (y-a)

^{2}= 9

x2 + y2 – 2ay + a2 – 9 = 0

Differentiating, we get

**Very Long Answer Type Questions (Score 6)**

Question 1.

i. Write the order and degree of the differential equation **(1) **

ii. Solve

**(2)**

iii. Solve

**(3)**

Answer:

i. Order = 2, Degree = 2

Question 2.

i. A spherical rain drop evaporates at a rate proportional to its surface area. If its radius is originally 3 mm

and after 1 hour it is reduced to 2 mm, find an expression for radius of rain drop at any time. **(3)**

ii. Solve

**(3)**

Answer:

Initially the radius is 3 mm, after 1 hour the radius is 2 mm. i.e., the radius changes 1 mm in 1 hour.

Volume is changing at a rate proportional to the surface area.

Question 3.

Given y dx – x dy + (log x) dx = 0

i. Express the given equation in the form **(2) **ii. Find the integrating factor.

**(1)**

iii. Solve the given differential equation.

**(3)**

Answer:

Question 4.

i. Write the equation of a circle having centre at (a, b) and radius ‘r’. ** (3) **

ii. By eliminating ‘a’ and ‘b’ form the differential equation corresponding to the family. **(3) **Answer:

i. Equation of the circle is:

(x – a)

^{2}+ (y – b)

^{2}= r

^{2}——-(1)

Question 5.

Solve

**(6)**

Answer:

Question 6.

Show that the general solution of the differential equation

is given by (x+y+1) = A (1 -x-y-2xy), where A is parameter. **(6)**

Answer:

Question 7.

Show that the differential equation

is homogeneous and solve it. **(6)**

Answer:

Question 8.

a. Solve **(3****)**

b. Solve

**(3)**

Answer:

Question 9.

Find the equation of the 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. **(6) **Answer:

We know that the slope of the tangent at any point (x, y) on the curve is given by

According to the given statement, we have

y

This is the equation of the family of solution curves of differential equation

(i). We have to find a particular memeber of the family which passes through (0,-2).

Substituting x = 0 and y = -2 in

(ii), we get =0+C ⇒ C – 2

Putting C =2 in (ii), we get

which is the equation of the required curve.

Question 10.

Solve the initial value problem

**(6)**

Answer:

**Edumate Questions & Answers**

Question 1.

i. Consider differential equation given below

Write the Oder and degree of th DE (if defined)

ii. From the differential equation of the equation y^{2}=a(b-x),a and b are arbitrary constants

Answer:

i. 4: degree is nit defined

Which is the differential equation.

Question 2.

i. Find the Differential equation satisfying the family of curves,y = ae^{3x} + be^{-2x} ,a and b are arbitary constants.

ii. Hence write the degree and order of the DE

Answer:

Question 3.

i. Choose the correct answer from the bracket

The solution of differential equation xdy+ydx=0 represents

(a) a rectangular hyperbola

(b) a parabola whose centre is origin.

(c) a straight line whose centre is origin

(d) a circle whose centre is origin.

ii. From the DE of the family of circles touching the DE of the family of circles touching the x-axis at origin.

Answer:

i. c. a straight line whose centre is origin.

ii. Let (0,a) be the centre of the circle.

Therefore the equation of the circle is x^{2} + (y-a^{2})^{2} = a^{2} ⇒ x^{2} = 2ay

Question 4.

Find aparticular solution satisfying the given condition

(x^{3} + x^{2} +x + 1) = 2x^{2} +x;y = 1 ,when x = 0

Answer:

Question 5.

Consider the DE (x + 2)(y+ 2) dx

i. Find the equation of the family of curves

ii. Find equation of the curve passing through the point (1,-1)

Answer:

Question 6.

Consider the differential equation

(i) Find

(ii) Solve the above differential equation

Answer:

Question 7.

i. Choose the correct answer from the bracket

The general solution of the DE

ii.

Solve the

Answer:

Question 8.

Choose the correct answer from the bracket determine the order and degree of the differential equation ,

(a). Fourth order , first degree

(b) Third order,first degree

(c) first order , fourth degree

(d) first order . third degree ]

(e)The population of a country doubles in 50 years . How many years will it be five times as much?Assume that the rate of increase is proportional to the number inhabitants. (hint:log2=0.6931;log5=1.6094)

Answer:

i. (a) Fourth order , first degree

Question 9.

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:

Let the rate of change of the volume of the balloon be k (where k is a constant)

At t = 3, r = 6;

⇒ 4π x 6^{3} = 3(k x 3+C)

⇒ 864;π =3(3A + 36π)

⇒ 3 A = 252;π => A = 84π

Substituting the values of k and C in equation (1), we get:

Thus, the radius of the ballon after t seconds is

**NCERT Questions & Answers**

Question 1.

Solve the differential equation:

Answer:

Question 2.

Solve the differential equation:

.

Answer:

Question 3.

It is known that, if the interest is compounded continously, the principal changes at the rate equal to the product of the rate of bank interest per annum and the principal.

i. Write a differential equation using the relation between the principal amount and the rate of interest.

ii. Solve the above differential equation, subject to the initial condition

P=P_{0} at t = 0

iii. How much will Rs.1000 be worth at 5% interest after 10 years?

(Given e°^{5} = 1.648)

Answer:

i. If P denotes the principal at any time t and the rate of interest be r% per an num compounded continously, then according to the law given in the problem.

Question 4.

i. If y = mx + c is a tangent to the circle circle x^{2}+y^{2} =1, show that

ii. Find the differncial equation of all stright lines touching circle x^{2}+y^{2} =1

Answer:

If y=m+c is a tangent ,perpendicular distance from centre(0,0) to mx+c must be the radius

Question 5.

Given

i. Express the differential equation as

function of

ii. Solve the differential equation using x = vy.

Answer:

Question 6.

Consider (1 + y^{2}) dx = (tan^{-1 }y – x) dy

i. Express the equation in the form

ii. Find the integrating factor.

iii. Solve the given equation.

Answer:

Question 7.

Consider

1. Find the general solution of the differential equation.

2. Find the equation of the curve that passes through (1, 2) and satisfies the differential equation.

Answer:

ii.Since the curve passes through (1,2) putting x = 1 and y = 2 we get c = 4

∴ Required equation is y(x^{2} + 1) = 4

Question 8.

i. Express the differential equation

in the form

ii. Find its integrating factor.

iii. Obtain the general solution

Answer:

Question 9.

Consider the differential equation

i. Write the order and degree.

ii. Show that xy = ae^{x} + be^{-x} + x^{2} is a solution of the given equation.

Answer:

i. Order = 2, Degree = 1

ii. xy = ae^{x} + be^{-x}. + x^{2}…………………… (1)

Differentiating w.r.t. x ,

Question 10.

Consider the differential equation (x^{2} – y^{2}dx+2xydy=0

i. Write the order and degree of differential equation.

ii. Show that the differential equation is homogeneous.

iii. Find the solution of the differential equation.

iv. Choose the correct solution from the following:

For y = 1 when x = 1

(x^{2} + y^{2} = -2x ; x^{2} + y^{2} = 2x ;

x^{2} + y^{2} – x = 0 ; x^{2} + y^{2} f x = 0)

Answer:

i. Order = 1, Degree = 1

Since each of the functions (y^{2} – x^{2}) and 2xy is a homogeneous function of degree 2, the given differential equation is therefore homogeneous.

Question 11.

Consider

i. Express the equation in the form

ii. Find the integrating factor.

iii. Solve the differential equation.

Answer:

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