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