## Chapter 9 - Differential Equations

#### Pages 382 - 383

Determine order and degree(if defined) of differential equation `(d^4y)/(dx^4) + sin(y^("')) = 0`

Determine order and degree(if defined) of differential equation y' + 5y = 0

Determine order and degree(if defined) of differential equation `((ds)/(dt))^4 + 3s (d^2s)/(dt^2) = 0`

Determine order and degree(if defined) of differential equation `(d^2y)/(dx^2)^2 + cos(dy/dx) = 0`

Determine order and degree(if defined) of differential equation `(d^2y)/(dx^2)` = cos 3x + sin 3x

Determine order and degree(if defined) of differential equation

( y′′′) + (y″)^{3} + (y′)^{4} + y^{5} = 0

Determine order and degree(if defined) of differential equation y′′′ + 2y″ + y′ = 0

Determine order and degree(if defined) of differential equation y′ + y = e^{x}

Determine order and degree(if defined) of differential equation y″ + (y′)^{2} + 2y = 0

Determine order and degree(if defined) of differential equation y″ + 2y′ + sin y = 0

The degree of the differential equation

`((d^2y)/(dx^2))^3 + ((dy)/(dx))^2 + sin ((dy)/(dx)) + 1 = 0`

**(A)** 3

**(B)** 2

**(C)** 1

**(D)** not defined

The order of the differential equation

`2x^2 (d^2y)/(dx^2) - 3 (dy)/(dx) + y = 0` is

**(A)** 2

**(B)** 1

**(C)** 0

**(D)** not defined

#### Page 385

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y = e^{x} + 1 : y″ – y′ = 0

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y = x^{2} + 2x + C : y′ – 2x – 2 = 0

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y = cos x + C : y′ + sin x = 0

`y sqrt(1 + x^2) : y' = (xy)/(1+x^2)`

y = Ax : xy′ = y (x ≠ 0)

y = x sin x : xy′ = `y + xsqrt(x^2 - y^2)` `(x != 0 and x > y, or x < -y)`

xy = log y + C : `y' = (y^2)/(1 - xy) (xy != 1)`

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:

y – cos y = x : (y sin y + cos y + x) y′ = y

x + y = tan^{–1}y : y^{2} y′ + y^{2} + 1 = 0

`y = sqrt(a^2 - x^2 ) x in (-a,a) : x + y dy/dx = 0(y != 0)`

The numbers of arbitrary constants in the general solution of a differential equation of fourth order are:

**(A)** 0

**(B)** 2

**(C)** 3

**(D)** 4

The numbers of arbitrary constants in the particular solution of a differential equation of third order are:

(A) 3

(B) 2

(C) 1

(D) 0

#### Page 391

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

`x/a + y/b = 1`

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y^{2} = a (b^{2} – x^{2})

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = a e^{3x} + b e^{– 2x}

y = e^{2x} (a + bx)

y = e^{x} (a cos x + b sin x)

Form the differential equation of the family of circles touching the *y*-axis at the origin.

Form the differential equation of the family of parabolas having vertex at origin and axis along positive *y*-axis.

Form the differential equation of the family of ellipses having foci on *y*-axis and centre at origin.

Form the differential equation of the family of hyperbolas having foci on *x*-axis and centre at origin.

Form the differential equation of the family of circles having centre on *y*-axis and radius 3 units.

Which of the following differential equations has y = c_{1} e^{x} + c_{2} e^{–x} as the general solution?

(A) `(d^2y)/(dx^2) + y = 0`

(B) `(d^2y)/(dx^2) - y = 0`

(C) `(d^2y)/(dx^2) + 1 = 0`

(D) `(d^2y)/(dx^2) - 1 = 0`

Which of the following differential equation has y = x as one of its particular solution?

A. `(d^2y)/(dx^2) - x^2 (dy)/(dx) + xy = x`

B. `(d^2y)/(dx^2) + x dy/dx + xy = x`

C. `(d^2y)/(dx^2) - x^2 dy/dx + xy = 0`

D. `(d^2y)/(dx^2) + x dy/dx + xy = 0`

#### Pages 395 - 397

For the differential equations find the general solution:

`dy/dx = (1 - cos x)/(1+cos x)`

For the differential equations find the general solution:

`dy/dx = sqrt(4-y^2) (-2 < y < 2)`

For the differential equations find the general solution:

`dy/dx + y = 1(y != 1)`

For the differential equations find the general solution:

sec^{2} x tan y dx + sec^{2} y tan x dy = 0

For the differential equations find the general solution:

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

For the differential equations find the general solution:

`dy/dx = (1+x^2)(1+y^2)`

For the differential equations find the general solution:

y log y dx – x dy = 0

For the differential equations find the general solution:

`x^5 dy/dx = - y^5`

For the differential equations find the general solution:

`dy/dx = sin^(-1) x`

For the differential equations find the general solution:

e^{x} tan y dx + (1 – e^{x}) sec^{2} y dy = 0

For each of the differential equations find a particular solution satisfying the given condition:

`(x^3 + x^2 + x + 1) dy/dx = 2x^2 + x; y = 1 `when x = 0`

For each of the differential equations find a particular solution satisfying the given condition:

`x(x^2 - 1) dy/dx = 1 , y = 0 " when x " = 2`

For each of the differential equations find a particular solution satisfying the given condition:

`cos (dx/dy) = a(a in R); y = 1 " when "x = 0`

For each of the differential equations find a particular solution satisfying the given condition:

`dy/dx` = y tan x; y = 1 when x = 0

Find the equation of a curve passing through the point (0, 0) and whose differential equation is y′ = e^{ x} sin x.

For the differential equation `xy(dy)/(dx) = (x + 2)(y + 2)` find the solution curve passing through the point (1, –1).

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.

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).

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.

In a bank, principal increases continuously at the rate of *r*% per year. Find the value of* r* if Rs 100 doubles itself in 10 years (log_{e}_{ }2 = 0.6931).

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)..

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?

The general solution of the differential equation `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

#### Pages 406 - 407

Show that the given differential equation is homogeneous and solve each of them

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

Show that the given differential equation is homogeneous and solve each of them

`y' = (x + y)/x`

Show that the given differential equation is homogeneous and solve each of them

(x – y) dy – (x + y) dx = 0

Show that the given differential equation is homogeneous and solve each of them

(x^{2} – y^{2}) dx + 2xy dy = 0

Show that the given differential equation is homogeneous and solve each of them

`x^2 dy/dx = x^2 - 2y^2 + xy`

Show that the given differential equation is homogeneous and solve each of them

`xdy - ydx = sqrt(x^2 + y^2) dx`

Show that the given differential equation is homogeneous and solve each of them

`{xcos(y/x) + ysin(y/x)}ydx = {ysin (y/x) - xcos(y/x)}xdy`

Show that the given differential equation is homogeneous and solve

`x dy/dx - y + x sin (y/x) = 0`

Show that the given differential equation is homogeneous and solve

`ydx + xlog(y/x)dy - 2xdy = 0`

Show that the given differential equation is homogeneous and solve

`(1+e^(x/y))dx + e^(x/y) (1 - x/y)dy = 0`

For the differential equations find the particular solution satisfying the given condition:

(x + y) dy + (x – y) dx = 0; y = 1 when x = 1

For the differential equations find the particular solution satisfying the given condition:

x^{2} dy + (xy + y^{2}) dx = 0; y = 1 when x = 1

For the differential equations find the particular solution satisfying the given condition:

`[xsin^2(y/x - y)] dx + xdy = 0; y = pi/4 when x = 1`

For the differential equations find the particular solution satisfying the given condition:

`dy/dx - y/x + cosec (y/x) = 0; y = 0 when x = 1`

For the differential equations find the particular solution satisfying the given condition:

`2xy + y^2 - 2x^2 dy/dx = 0; y = 2 " when x " = 1`

A homogeneous differential equation of the from `dx/dy = h(x/y)` can be solved by making the substitution

**A.** *y* = *vx*

**B.** *v* = *yx*

**C.** *x *= *vy*

**D. ***x* =* v*

Which of the following is a homogeneous differential equation?

(A) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0

(B) (xy) dx – (x^{3} + y^{3}) dy = 0

(C) (x^{3} + 2y^{2}) dx + 2xy dy = 0

(D) y^{2} dx + (x^{2} – xy – y^{2}) dy = 0

#### Pages 413 - 414

For the differential equations find the general solution:

`dy/dx + 2y = sin x`

For the differential equations find the general solution:

`dy/dx + 3y = e^(-2x)`

For the differential equations find the general solution:

`dy/dx + y/x = x^2`

For the differential equations find the general solution:

`dy/dx + secxy = tan x (0 <= x < pi/2)`

For the differential equations find the general solution:

`cos^2 x dy/dx + y = tan x(0 <= x < pi/2)`

For the differential equations find the general solution:

`x dy/dx + 2y= x^2 log x`

For the differential equations find the general solution:

`x log x dy/dx + y= 2/x log x`

For the differential equations find the general solution: (1 + x^{2}) dy + 2xy dx = cot x dx (x ≠ 0)

For the differential equations find the general solution:

`x dy/dx + y - x + xycot x = 0(x != 0)`

For the differential equations find the general solution:

`(x + y) dy/dx = 1`

For the differential equations find the general solution:

y dx + (x – y^{2}) dy = 0

For the differential equations find the general solution:

`(x + 3y^2) dy/dx = y(y > 0)`

For the differential equations given find a particular solution satisfying the given condition:

`dy/dx + 2y tan x = sin x; y = 0 " when x " = pi/3`

For the differential equations given find a particular solution satisfying the given condition:

`(1 + x^2)dy/dx + 2xy = 1/(1 + x^2); y = 0 " when x " =1`

For the differential equations given find a particular solution satisfying the given condition:

`dy/dx - 3ycotx = sin 2x; y = 2 " when x "= pi/2`

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (*x*, *y*) is equal to the sum of the coordinates of the point.

Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

The Integrating Factor of the differential equation `dy/dx - y = 2x^2` is

**A.** *e*^{–}^{x}

**B.** *e*^{–}^{y}

**C.** 1/x

**D. ***x*

The integrating factor of the differential equation.

`(1 - y^2) dx/dy + yx = ay(-1 < < 1)` is

#### Pages 419 - 421

For differential equations given below, indicate its order and degree (if defined).

`(d^2y)/dx^2 + 5x(dy/dx)^2 - 6y = log x`

For differential equations given below, indicate its order and degree (if defined).

`((dy)/(dx))^3 -4(dy/dx)^2 + 7y = sin x`

For differential equations given below, indicate its order and degree (if defined).

`(d^4y)/dx^4 - sin ((d^3y)/(dx^3)) = 0`

For given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

y = a e^{x} + b e^{–x} + x^{2 : `x (d^2y)/(dx62) + 2 dy/dx - xy + x^2 - 2 = 0`}

For given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

`y = e^x (acos x + b sin x) : (d^2y)/(dx^2) - 2 dy/dx + 2y = 0`

For given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

`y = xsin 3x : (d^2y)/(dx^2) + 9y - 6 cos 3x = 0`

`x^2 = 2y^2 log y : (x^2 + y^2) dy/dx - xy = 0`

Form the differential equation representing the family of curves given by (x – a)^{2} + 2y^{2} = a^{2}, where a is an arbitrary constant.

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Prove that x^{2} – y^{2} = c (x^{2} + y^{2})^{2} is the general solution of differential equation (x^{3} – 3x y2) dx = (y^{3} – 3x^{2}y) dy, where c is a parameter.

Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0`

Show that the general solution of the differential equation `dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0` is given by (*x* + *y *+ 1) = *A *(1 – *x *– *y* – 2*xy*), where *A *is parameter

Find the equation of the curve passing through the point `(0,pi/4)`, whose differential equation is sin x cos y dx + cos x sin y dy = 0.

Find the particular solution of the differential equation (1 + e^{2x}) dy + (1 + y^{2}) ex dx = 0, given that y = 1 when x = 0.

Solve the differential equation `ye^(x/y) dx = (xe^(x/y) + y^2)dy (y != 0)`

Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)

Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0)`

Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that *y* = 0 when `x = pi/2`

Find a particular solution of the differential equation`(x + 1) dy/dx = 2e^(-y) - 1`, given that *y* = 0 when *x* = 0

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

The general solution of the differential equation `(ydx - xdy)/y = 0`

**A.** *xy *= C

**B.** *x *= C*y*^{2}

**C.** *y *= C*x*

**D. ***y* = C*x*^{2}

The general solution of a differential equation of the type `dx/dy + P_1 x = Q` is

(A) `y e^(intP_1 dy) = int(Q_1 e^(intP_1 dy)) dy + C`

(B) `y . e^(intP_1 dx) = int(Q_1 e^(intP_1 dx)) dx + C`

(C) `x e^(intP_1 dy) = int(Q_1 e^(intP_1 dy)) dy + C`

(D) `xe^(intP_1 dx) = int(Q_1 e^(intP_1 dx)) dx + C`

The general solution of the differential equation e^{x} dy + (y e^{x} + 2x) dx = 0 is

**A.** *xe*^{y} + *x*^{2} = C

**B.** *xe*^{y} + *y*^{2} = C

**C.** *ye*^{x} + *x*^{2} = C

**D. ***ye*^{y}^{ }+ *x*^{2} = C