KC Sinha Solution Class 12 Chapter 23 Differential Equations Exercise 23.1

 Exercise 23.1

Question 1

$\frac{d^{2} y}{d x^{2}}+4 x=0$

Sol :

Order=2

Degree=1

Linear Differential Equation

Question 2

$3 \frac{d^{2} y}{d x^{2}}-5\left(\frac{dy}{dx}\right)^{3}+2 y=0$

Sol :

Order=2

Degree=1

Non Linear Differential Equation

Question 3

$\left[1+\left(\frac{d y}{dx}\right)^{2}\right]^{\frac{3}{2}}=\frac{d^2 y}{d x^{2}}$

Sol :

Order=2

Degree=2

Non-Linear Differential Equation

$\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3/2}=\frac{d^2 y}{dx^{2}}$

Squaring both sides

$\left[\left(1+\left(\frac{dy}{dx}\right)^{2}\right]^{3/ 2}\right]^{2}=\left(\frac{d^2 {y}}{dx^{2}}\right)^{2}$

$\left(1+\left(\frac{d y}{d y}\right)^{2}\right)^{3}=\left(\frac{d^{2} y}{d x^{2}}\right)^{2}$

Question 4

$y=x \frac{d y}{d x}+a \sqrt{1+\left(\frac{d y}{d x}\right)^{3}}$

Sol :

Order=1

Degree=3

Non-Linear Differential Equation

$y=x \frac{d y}{d x}+a \sqrt{1+\left(\frac{d y}{dx}\right)^{3}}$

$y-x \frac{d y}{d x}=a \sqrt{1+\left(\frac{d y}{dx}\right)^{3}}$

Squaring both sides

$\left(y-x \frac{d y}{d x}\right)^{2}=a\left(1+\left(\frac{dy}{dx}\right)^{3}\right)$

Question 5

$\frac{d^{3} y}{d x^{3}}+3 \frac{d^{2} y}{dx^{2}}+3 \frac{d y}{dx}+y=0$

Sol :

Order=3

Degree=1

Question 6

$x^{2}\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+y\left(\frac{d y}{dx}\right)^{4}+y^{4}=0$

Sol :

Order=2

Degree=3

Question 7

$\frac{d^{2} y}{d x^{2}}=\cos 3x+\sin 3 x$

Sol :

Order=2

Degree=1

Question 8

$2 x^{2} \frac{d^2y}{d x^{2}}-3 \frac{d y}{dx}+y=0$

Sol :

Order=2

Degree=1

Question 9

y¹+5y=0

Sol :

Order=2

Degree=1

Question 10

y¹¹¹+2y¹¹+y¹=0

Sol :

Order=3

Degree=1

Question 14

(y¹¹)²+(y¹¹)³+(y¹)⁴+y⁵=0

Sol :

Order=3

Degree=2

Question 15

$\frac{d y}{d x}=e^{x}$

Sol :

Order=1

Degree=1

Question 18

(x+y-3)dx+(x²+3x+y)dy=0

Sol :

(x²+3x+y)dy=-(x+y-3)dx

(x²+3x+y)$\frac{dy}{dx}$=-(x+y-3)

Question 19

$\frac{d^{2} y}{d x^{2}}=3 \sqrt{1+\left(\frac{d y}{d x}\right)^{2}}$

Sol :

$\left(\frac{d^{2} y}{d x^{2}}\right)^{3}=1+\left(\frac{d y}{dx}\right)^{2}$

Order=2

Degree=3

Question 20

$\sqrt[3]{\frac{d^{2} y}{d x^{2}}}=\sqrt{\frac{d y}{dx}}$

Sol :

$\left(\frac{d ^2y}{d x^2}\right)=\left(\frac{(d y)^{\frac{1}{2}}}{dx}\right)^{3}$

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

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

Question 21

$\frac{d^{4} y}{d x^{4}}=\left[c+\left(\frac{d y}{d x}\right)^{2}\right]^{3 / 2}$

Sol :

$\left(\frac{d^{4} y}{d x^{4}}\right)^{2}=\left[c+\left(\frac{d y}{d x}\right)^{2}\right]^3$

Order=4

Degree=2

Question 22

$s^{2} \frac{d^{2} t}{d s^{2}}+s t \frac{d t}{d s}=s$

Sol :

Order=2

Degree=1

Question 24

$5 \frac{d^{2} y}{d x^2}=\left(1+\left(\frac{d y}{dx}\right)^{2}\right]^{\frac{3}{2}}$

Sol :

Order=1

Degree=2

Question 32

$\left(\frac{d ^2y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d y}\right)=0$

Sol :

Order=2

$\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{dx}\right)=0$

$\left(\frac{d ^2y}{d x^{2}}\right)^{2}=-\cos\left(\frac{d y}{dx}\right)$

$-\left(\frac{d^{2} x}{d x^{2}}\right)^{2}=\cos \left(\frac{dy}{d x}\right)$

$\cos^{-1}\left\{-\left(\frac{d^{2} y}{d x^{2}}\right)^{2}\right\}=\frac{d y}{d x}$

Degree=Not defined

Question 33

$\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d y}\right)^{2}+\sin \left(\frac{d y}{dx}\right)+1=0$

Sol :

Order=2

Degree=Not defined

Question 34

y^m+y²+$e^y^1$=0

Sol :

Order=3

Degree=Not defined

Question 36

$1+\frac{d y}{d x}=e^{\frac{dy}{dx}}$

Question 37

$1+x=e^{\frac{d y}{dx}}$

Sol :

$\log(1+x)=\log e^{\left(\frac{dy}{dx}\right)}$

$\log ()1+x=\frac{d y}{d x} \log e$

$\lg (1+x)=\frac{d y}{dx}$

Order=1

Degree=1

Question 38

$\log \left(1+\frac{d y}{dx}\right)=x+y$

Sol :

$1+\frac{dy}{dx}=e^{x+y}$

Order=1

Degree=1

Question 45

$x-e^{\frac{dy}{dx}}=0$

Sol :

$x=e^{\frac{dy}{dx}}$

$\log x=\log e^{\frac{dy}{dx}}$

$\log x=\frac{dy}{dx}\log e$

$\log x=\frac{dy}{dx}$