## Differential Equations

Extra questions of chapter 09 Differential Equations, class XII  with answers and  hints to the difficult questions, strictly according to the CBSE Board syllabus. Important and useful math. assignment for the students of class XII

## MATHEMATICS ASSIGNMENT OF EXTRA QUESTION

STRICTLY ACCORDING TO THE PREVIOUS CBSE SAMPLE QUESTION PAPERS AND CBSE BOARD PAPERS
Question: 1 If m and n, respectively, are the order and the degree of the differential equation  $\frac{d}{dx}\left [ \frac{dy}{dx} \right ]^{4}=0$ , then find m + n

Ans: 3
Solution Hint: Solve the above differential we get $4\left ( \frac{dy}{dx} \right )^{3}\frac{d^{2}y}{dx^{2}}=0$
Here m = 2 and n = 1 so m + n = 3
Question: 2 Write the order of the differential equation:

$log\left ( \frac{d^{2}y}{dx^{2}} \right )=\left ( \frac{dy}{dx} \right )^{3}+x$

Question: 3 Find the general solution of the differential equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0
Question: 4 Solve the differential equation: ydx + (x – y2)dy = 0

Answer: $xy=\frac{y^{3}}{3}+C$

Solution Hint: Reduce it to LDE and then solve it
Question: 5 Solve the differential equation:  $xdy-ydx=\sqrt{x^{2}+y^{2}}dx$

Answer: $y+\sqrt{x^{2}+y^{2}}=Cx^{2}$
Solution Hint: Reduce it to HDE and then solve it
Question: 6
Find the general solution of the  differential equation $x\frac{dy}{dx}=y-x\: sin\left ( \frac{y}{x} \right )$

Answer: $\left [ cosec\left ( \frac{y}{x} \right )-cot\left ( \frac{y}{x} \right ) \right ]x=C$

Solution Hint: Reduce the given equation into HDE and then solve it.
Question: 7
Find the particular solution of the following differential equation, given that y = 0 when x = π/4
$\frac{dy}{dx}+ycotx=\frac{2}{1+sinx}$

Answer:  $y\: sinx=2\left [ x+tan\left ( \frac{\pi }{4}-\frac{x}{2} \right ) \right ]-\left ( \frac{\pi }{2}+2tan\frac{\pi }{8} \right )$

Solution Hint: Convert the given differential equation in LDE

Find Integrating Factor (IF) = sin x

Multiply by integrating factor on both side and then integrating we get

$y\: sinx=2\left [ x+tan\left ( \frac{\pi }{4}-\frac{x}{2} \right ) \right ]+C$
Now putting y = 0 when x = π/4 and find C,  the value of  C is given by

$C=-\left ( \frac{\pi }{2}+2tan\frac{\pi }{8} \right )$

Now require particular solution is

$y\: sinx=2\left [ x+tan\left ( \frac{\pi }{4}-\frac{x}{2} \right ) \right ]-\left ( \frac{\pi }{2}+2tan\frac{\pi }{8} \right )$
Question: 8
Write the general solution of differential equation   $\frac{dy}{dx}=e^{x+y}$

Solution Hint: Given differential equation can be written as
$\frac{dy}{dx}=e^{x}.e^{y}$
Separating the denominator we get  $e^{-y}dy=e^{x}dx$

Integrating on both side we get:
$\int e^{-y}dy=\int e^{x}dx\Rightarrow -e^{-y}+C=e^{x}$

⇒ ex +e-y = C

Question : 9  Find the particular solution of the following differential equation
$cosy\: dx+(1+2e^{-x})siny\: dy=0;\: \: \: y(0)=\frac{\pi }{4}$
Answer: $e^{x}+2=3\sqrt{2}\: cosy$

Solution Hint: Use variable separable and integrating on both side we get

$\int \frac{e^{x}}{2+e^{x}}dx=\int -\frac{siny}{cosy}dy$

$log(e^{x}+2)=log\: cosy+logC$

$e^{x}+2=Ccosy$

Now putting x = 0 and y = π/4 we get
C = 3√2
Now particular solution is given by

$e^{x}+2=3\sqrt{2}\: cosy$
Question:10 Find the general solution of the differential equation:

$\frac{dx}{dy}=\frac{y\: tany-x\: tany-xy}{y\: tany}$

Answer:   xy sin y = sin y - y cos y + C

Solution Hint:
Separating the variables we get LDE shown below

$\frac{dx}{dy}+\left ( \frac{1}{y}+\frac{1}{tany} \right )x=1$

$IF=e^{\left ( \frac{1}{y}+\frac{1}{tany} \right )dy}=e^{logy+logsiny}=e^{logysiny}$

IF = y sin y

Multiplying on both side by IF and then integrating we get

x.y sin y = sin y - y cos y + C
Question: 11
Find the particular solution of the differential equation : yexdx=(y+ 2xey)dy ,  At  y(0) = 1

Answer: x = - y2e-y + y2/e

Solution Hint:  Given equation can be written as

$\frac{dx}{dy}+\left ( -\frac{2}{y} \right )x=\frac{y^{2}}{e^{y}}$

It is Linear Differential Equation in x

Integrating Factor = 1/y2.

Multiply on both side by I.F. and then integrating we get

x = - y2e-y + Cy2.

Now putting x = 0 and y = 1, we get,  C = 1/e

So particular solution is:  x = - y2e-y + y2/e

Question 12
Show that (x - y)dy = (x + 2y)dx is a homogenous differential equation. Also, find the general solution of the given differential equation.

Answer:    $log(y^{2}+xy+x^{2})-2\sqrt{3}tan^{-1}\left ( \frac{2y+x}{\sqrt{3}x} \right )=C$

Solution Hint:  Given differential equation can be written as

$\frac{dy}{dx}=\frac{x+2y}{x-y}=\frac{1+\frac{2y}{x}}{1-\frac{y}{x}}$

It is a Homogeneous differential equation. So solving it by making the substitution y/x = v and then integrating we get general solution

$log(y^{2}+xy+x^{2})-2\sqrt{3}tan^{-1}\left ( \frac{2y+x}{\sqrt{3}x} \right )=C$
Question 13

Find the value of (2a - 3b), if a and b represent respectively the order and the degree of the differential equation.

$x\left [ y\left ( \frac{d^{2}y}{dx^{2}} \right )^{3}+x\left ( \frac{dy}{dx} \right )^{2}-\frac{y}{x}\left ( \frac{dy}{dx} \right ) \right ]=0$

Order = 2 and Degree = 3

⇒ a = 2 and b = 3

2a - 3b = 2 × 2 - 3 × 3 = 4 - 9 = - 5

Question 14
Solve the differential equation  (ex + 1)y dy = ex(y + 1)dx

Answer: log (ex + 1) = y - log(y + 1) + C

Question 15

Solve the following homogeneous differential equation :  $x\frac{dy}{dx}=x+y$

Answer: y = x ( log |x| + C )

Question 16

Solve the differential equation  $x\frac{dy}{dx}=y-xtan\left ( \frac{y}{x} \right )$

Solution Hint

Write this equation in the standard form given equation reduces to HDE

Now putting y/x = v and integrating on both side we get

$\int cotv\: dv=-\int \frac{dx}{x}$

log sin v = - log x + log C

$sin\left ( \frac{y}{x} \right )=\frac{C}{x}\: \: \Rightarrow \: \: xsin\left ( \frac{y}{x} \right )=C$

Question: 17  Solve the following differential equation

$\frac{dy}{dx}=-\left [ \frac{x+ycosx}{1+sinx} \right ]$

Solution Hint:  Simplify this equation we get

$\frac{dy}{dx}+\frac{y\: cosx}{1+sinx}=\frac{-x}{1+sinx}$

This is Linear Differential Equation

Finding IF and then integrating on both sides we get

$y(1+sinx)=-\frac{x^{2}}{2}+C$

Question: 18: Find the particular solution of the following differential equation.

ex tan y dx + (2 - ex) sec2y dy = 0,  given that y = π/4 when x = 0

Solution Hint: Separate the variables we get

$\frac{e^{x}}{e^{x}-2}dx=\frac{sec^{2}y}{tany}dy$

Integrating on both side we get   ex  – 2 = C tan y

Now putting y = π/4 and x = 0 we get C = -1

Required particular solution is

y =  tan-1(2 + ex )

Question: 19: Solve the followings differential equation$x\: sin\left ( \frac{y}{x} \right )\frac{dy}{dx}+x-y\: sin\left ( \frac{y}{x} \right )=0$ ,  given  that x = 1 at y = π/2

Solution Hint:

Reduce this equation into HDE and then putting y/x = v we get

$siv\: dv=-\frac{1}{x}dx$

Now integrating on both side and putting v = y/x we get

$-cos\left ( \frac{y}{x} \right )=-log|x|+C$

Now putting x = 1 and y = π/2 we get C = 0

Required solution of the given differential equation is

$cos\left ( \frac{y}{x} \right )=log|x|$

Question: 20: Solve the following differential equations

$(x^{3}+x^{2}+x+1)\frac{dy}{dx}=2x^{2}+x$ ,  given that x = 0 at y = 1

Solution Hint:

Simplify the differential equation we get

$dy=\frac{2x^{2}+x}{(x^{2}+1)(x+1)}dx$
Now integrating on both sides and by using the partial fraction we get
$y=\frac{1}{2}\int \frac{1}{x+1}dx+\int \frac{\frac{3}{2}x-\frac{1}{2}}{x^{2}+1}dx$
$y=\frac{1}{2}log(x+1)+\frac{3}{2}log(x^{2}+1)-\frac{1}{2}tan^{-1}x+C$
When x = 0, y = 1 we get C = 1

$y=\frac{1}{2}log(x+1)+\frac{3}{2}log(x^{2}+1)-\frac{1}{2}tan^{-1}x+1$

$y=\frac{1}{4}log\left [ (x+1)^{2}(x^{2}+1)^{3} \right ]-\frac{1}{2}tan^{-1}x+1$
Question 21
Form the differential equation of all circles which is touching the x-axis at the origin.

Solution Hint
Equation of circle with centre C(0, r) and radius r is given by
(x - 0)(y - 0)r2

x+ y2 = 2ry ........(i)

Differentiating w.r.t. x we get

2x + 2yy' = 2ry'

From this equation find the value of r and putting this value in equation (i) we get

(x+ y2)y' = 2y(x + yy’)

This is the required differential equation

Question 22

Find the differential equation of the family of curves  y= 4ax

Ans:  2xyy' - y2 = 0

Question 23

Find the solution of the differential equation  $\frac{dy}{dx} = x^{3}e^{-2y}$

Solution Hint:  Separate the variables and the integrating we get:

2e2y = x4 + C1

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