12th NCERT Differential Equation Exercise 9.2 Number of questions 12
Hi
In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

Question (1)

y = ex + 1 : y'' - y' = 0

Solution

y = ex + 1
differentiate w.r.t to x
y' = ex
differentiate w.r.t to x
y'' = ex
y'' = y'
∴ y'' - y' = 0
It is given differential equation so, y= ex+1 is solution of gives differential equation

Question (2)

y=x2 + 2x + C : y' -2x - 2 = 0

Solution

y = x2 + 2x + c
differentiate w.r.t to x
$\frac{{dy}}{{dx}} = 2x + 2$ ∴ y' - 2x -2 = 0

Question (3)

y=cosx + C : y' + sinx = 0

Solution

Differentiate w.r.t x
$\frac{{dy}}{{dx}} = - \sin x$ y' + sinx =0

Question (4)

$y = \sqrt {1 + {x^2}} \;:\;y' = \frac{{xy}}{{1 + {x^2}}}$

Solution

Differentiate w.r.t x
$\frac{{dy}}{{dx}} = \frac{1}{{2\sqrt {1 + {x^2}} }} \cdot \frac{d}{{dx}}\left( {1 + {x^2}} \right)$ $y' = \frac{1}{{\require{cancel}\cancel{2}\sqrt {1 + {x^2}} }} \cdot \cancel{2}x$ $y' = \frac{x}{{\sqrt {1 + {x^2}} }}$ $y' = \frac{{x\sqrt {1 + {x^2}} }}{{1 + {x^2}}}$ $y' = \frac{{xy}}{{1 + {x^2}}}$

Question (5)

y = Ax : xy' = y (x ≠ 0)

Solution

Differentiate w.r.t x
$\frac{{dy}}{{dx}} = A$ y'=A
multiply by x xy' = Ax
xy' = y

Question (6)

$\begin{array}{l}y = x\sin x\;:\;xy' = y + x\sqrt {{x^2} - {y^2}} \\\quad \quad \quad \quad \;(x \ne 0{\kern 1pt} \,and\,x > y{\kern 1pt} \,or\,x < - y)\end{array}$

Solution

Differentiate w.r.t x
$y' = x\frac{d}{{dx}}\sin x + \sin x\frac{d}{{dx}}x$ y'= xcosx + sinx
multiply by x
∴ xy' = x2cosx + xsinx xy' = x2cosx + y xy'=x(xcosx) + y $xy' = x\sqrt {{x^2}{{\cos }^2}x} + y$ $xy' = x\sqrt {{x^2}\left( {1 - {{\sin }^2}x} \right)} + y$ $xy' = x\sqrt {{x^2} - {x^2}{{\sin }^2}x} + y$ $xy' = x\sqrt {{x^2} - {y^2}} + y$

Question (7)

$xy = \log y + C\;\quad :\quad y' = \frac{{{y^2}}}{{1 - xy}}\left( {xy \ne 1} \right)$

Solution

Differentiate w.r.t x
$x\frac{{dy}}{{dx}} + y\left( 1 \right) = \frac{1}{y}\frac{{dy}}{{dx}}$ $x\frac{{dy}}{{dx}} - \frac{1}{y}\frac{{dy}}{{dx}} = - y$ $\left( {\frac{{xy - 1}}{y}} \right)\frac{{dy}}{{dx}} = - y$ $y' = \frac{{ - {y^2}}}{{xy - 1}}$ $y' = \frac{{{y^2}}}{{1 - xy}}$

Question (8)

y-cosy = x : (ysiny + cosy +x)y' = y

Solution

Differentiate w.r.t x
$y' - \sin y \cdot \frac{{dy}}{{dx}} = 1$ $y' - \sin y \cdot y' = 1$ multiply by y yy' + ysiny. y' = y
y'(y + ysiny) = y y'(x+cosy + ysiny) = y ∴ (ysiny + cosy + x) y' = y

Question (9)

x + y = tan-1y : y2y' + y2 + 1 = 0

Solution

Differentiate w.r.t x
$1 + \frac{{dy}}{{dx}} = \frac{1}{{1 + {y^2}}} \cdot \frac{{dy}}{{dx}}$ $1 + y' = \frac{{y'}}{{1 + {y^2}}}$ (1+y2) (1+y') = y'
1+ \cancel{y'}+y2+y2y' = \cancel{y'} y2y'+ y2 + 1 = 0

Question (10)

$y = \sqrt {{a^2} - {x^2}} \;x \in \left( { - a,a} \right):x + y\frac{{dy}}{{dx}} = 0(y \ne 0)$

Solution

Differentiate w.r.t x
$\frac{{dy}}{{dx}} = \frac{1}{{\cancel{2}\sqrt {{a^2} - {x^2}} }} \times \left( { - \cancel{2}x} \right)$ $y' = \frac{{ - x}}{y}$ yy' = -x
x + yy' = 0

Question (11)

The number of arbitrary constants in the general solution of a differential equation of fourth order are
(A) 0   (B) 2   (C) 3   (D) 4

Solution

Number of arbitrary constant of 4th order are 4
as number of arbitrary constant = degree

Question (12)

The number of arbitrary constants in the particular solution of a differential equation of third order are
(A) 3   (B) 2   (C) 1   (D) 0

Solution

Number of arbitrary constant in perticular solution = 0