\displaystyle \text{Determine the order and degree of each of the following differential } \\ \text{equations. State also whether they are linear or non-linear:}

\displaystyle \textbf{Question 1: }~\frac{d^{3}x}{dt^{3}}+\frac{d^{2}x}{dt^{2}}+\left(\frac{dx}{dt}\right)^{2}=e^{t}.
\displaystyle \text{Answer:}
\displaystyle \text{Here, the order of the differential equation is }3\text{ since the highest order derivative is of order } \\ 3\text{, and the degree is }1\text{ since the highest order derivative appears to the first power.}
\displaystyle \text{The differential equation is non\text{-}linear because the derivative }\frac{dx}{dt} \\ \text{ appears with power }2\text{, which is greater than }1.

\displaystyle \textbf{Question 2: }~\frac{d^{2}y}{dx^{2}}+4y=0.
\displaystyle \text{Answer:}
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative is of order } \\ 2\text{, and the degree is }1\text{ since the highest order derivative appears to the first power.}
\displaystyle \text{The given differential equation is linear.}

\displaystyle \textbf{Question 3: }~\left(\frac{dy}{dx}\right)^{2}+\frac{1}{dy/dx}=2.
\displaystyle \text{Answer:}
\displaystyle \left(\frac{dy}{dx}\right)^2+\frac{1}{\left(\frac{dy}{dx}\right)}=2
\displaystyle \Rightarrow \left(\frac{dy}{dx}\right)^3+1=2\frac{dy}{dx}
\displaystyle \Rightarrow \left(\frac{dy}{dx}\right)^3-2\frac{dy}{dx}+1=0
\displaystyle \text{Here, the order of the differential equation is }1\text{ since the highest order derivative present is } \\ \frac{dy}{dx}\text{, and the degree is }3\text{ since this derivative appears with highest power }3.
\displaystyle \text{The differential equation is non\text{-}linear because the derivative } \\ \frac{dy}{dx}\text{ appears with power }3\text{, which is greater than }1.

\displaystyle \textbf{Question 4: }~\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}=\left(c\frac{d^{2}y}{dx^{2}}\right)^{1/3}.
\displaystyle \text{Answer:}
\displaystyle \sqrt{1+\left(\frac{dy}{dx}\right)^2}=\left(c\frac{d^2y}{dx^2}\right)^{\frac{1}{3}}
\displaystyle \text{Squaring both sides, we get}
\displaystyle 1+\left(\frac{dy}{dx}\right)^2=\left(c\frac{d^2y}{dx^2}\right)^{\frac{2}{3}}
\displaystyle \text{Taking cubes of both sides, we get}
\displaystyle \left(c\frac{d^2y}{dx^2}\right)^2=\left[1+\left(\frac{dy}{dx}\right)^2\right]^3
\displaystyle \Rightarrow c^2\left(\frac{d^2y}{dx^2}\right)^2=1+3\left(\frac{dy}{dx}\right)^2+3\left(\frac{dy}{dx}\right)^4+\left(\frac{dy}{dx}\right)^6
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2y}{dx^2}\text{, and the degree is }2\text{ since it appears with highest power }2.
\displaystyle \text{The differential equation is non\text{-}linear because the degree of the equation is greater than }1.

\displaystyle \textbf{Question 5: }~\frac{d^{2}y}{dx^{2}}+\left(\frac{dy}{dx}\right)^{2}+xy=0.
\displaystyle \text{Answer:}
\displaystyle \frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2+xy=0
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2y}{dx^2}\text{, and the degree is }1\text{ since this derivative appears to the first power.}
\displaystyle \text{The differential equation is non\text{-}linear because the term }\left(\frac{dy}{dx}\right)^2 \\ \text{ involves the derivative raised to a power greater than }1.

\displaystyle \textbf{Question 6: }~\sqrt[3]{\frac{d^{2}y}{dx^{2}}}=\sqrt{\frac{dy}{dx}}.
\displaystyle \text{Answer:}
\displaystyle \sqrt[3]{\frac{d^2y}{dx^2}}=\sqrt{\frac{dy}{dx}}
\displaystyle \Rightarrow \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{3}}=\left(\frac{dy}{dx}\right)^{\frac{1}{2}}
\displaystyle \text{Taking cubes of both sides, we get}
\displaystyle \frac{d^2y}{dx^2}=\left(\frac{dy}{dx}\right)^{\frac{3}{2}}
\displaystyle \text{Squaring both sides, we get}
\displaystyle \left(\frac{d^2y}{dx^2}\right)^2=\left(\frac{dy}{dx}\right)^3
\displaystyle \Rightarrow \left(\frac{d^2y}{dx^2}\right)^2-\left(\frac{dy}{dx}\right)^3=0
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2y}{dx^2}\text{, and the degree is }2\text{ since this derivative appears with highest power }2.
\displaystyle \text{Thus, the differential equation is non\text{-}linear because its degree is } \\ 2\text{, which is greater than }1.

\displaystyle \textbf{Question 7: }~\frac{d^{4}y}{dx^{4}}=\left\{c+\left(\frac{dy}{dx}\right)^{2}\right\}^{3/2}.
\displaystyle \text{Answer:}
\displaystyle \frac{d^4y}{dx^4}=\left[c+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}
\displaystyle \text{Squaring both sides, we get}
\displaystyle \left(\frac{d^4y}{dx^4}\right)^2=\left[c+\left(\frac{dy}{dx}\right)^2\right]^3
\displaystyle \Rightarrow \left(\frac{d^4y}{dx^4}\right)^2=c^3+3c^2\left(\frac{dy}{dx}\right)^2+3c\left(\frac{dy}{dx}\right)^4+\left(\frac{dy}{dx}\right)^6
\displaystyle \text{Here, the order of the differential equation is }4\text{ since the highest order derivative present is } \\ \frac{d^4y}{dx^4}\text{, and the degree is }2\text{ since this derivative appears with highest power }2.
\displaystyle \text{Thus, the differential equation is non\text{-}linear because its degree is }2\text{, which is greater than }1.

\displaystyle \textbf{Question 8: }~x+\left(\frac{dy}{dx}\right)=\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}.
\displaystyle \text{Answer:}
\displaystyle x+\frac{dy}{dx}=\sqrt{1+\left(\frac{dy}{dx}\right)^2}
\displaystyle \Rightarrow x+\frac{dy}{dx}=\left(1+\left(\frac{dy}{dx}\right)^2\right)^{\frac{1}{2}}
\displaystyle \text{Squaring both sides, we get}
\displaystyle \left(x+\frac{dy}{dx}\right)^2=1+\left(\frac{dy}{dx}\right)^2
\displaystyle \Rightarrow x^2+2x\frac{dy}{dx}+\left(\frac{dy}{dx}\right)^2=1+\left(\frac{dy}{dx}\right)^2
\displaystyle \Rightarrow 2x\frac{dy}{dx}+x^2=1
\displaystyle \text{Here, the order of the differential equation is }1\text{ since the highest order derivative present is } \\ \frac{dy}{dx}\text{, and the degree is }1\text{ since it appears to the first power.}
\displaystyle \text{Hence, the differential equation is linear.}

\displaystyle \textbf{Question 9: }~y\frac{d^{2}x}{dy^{2}}=y^{2}+1.
\displaystyle \text{Answer:}
\displaystyle y\frac{d^2x}{dy^2}=y^2+1
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2x}{dy^2}\text{, and the degree is }1\text{ since it appears to the first power.}
\displaystyle \text{The given differential equation is linear.}

\displaystyle \textbf{Question 10: }~s^{2}\frac{d^{2}t}{ds^{2}}+st\frac{dt}{ds}=s.
\displaystyle \text{Answer:}
\displaystyle s^2\frac{d^2t}{ds^2}+s\frac{dt}{ds}=s
\displaystyle \Rightarrow s\frac{d^2t}{ds^2}+\frac{dt}{ds}=1
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2t}{ds^2}\text{, and the degree is }1\text{ since it appears to the first power.}
\displaystyle \text{The given differential equation is linear.}

\displaystyle \textbf{Question 11: }~x^{2}\left(\frac{d^{2}y}{dx^{2}}\right)^{3}+y\left(\frac{dy}{dx}\right)^{4}+y^{4}=0.
\displaystyle \text{Answer:}
\displaystyle x^2\left(\frac{d^2y}{dx^2}\right)^3+y\left(\frac{dy}{dx}\right)^4+y^4=0
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2y}{dx^2}\text{, and the degree is }3\text{ since this derivative appears with highest power }3.
\displaystyle \text{The differential equation is non\text{-}linear because its degree is greater than }1.

\displaystyle \textbf{Question 12: }~\frac{d^{3}y}{dx^{3}}+\left(\frac{d^{2}y}{dx^{2}}\right)^{3}+\frac{dy}{dx}+4y=\sin x.
\displaystyle \text{Answer:}
\displaystyle \frac{d^3y}{dx^3}+\left(\frac{d^2y}{dx^2}\right)^3+\frac{dy}{dx}+4y=\sin x
\displaystyle \text{Here, the order of the differential equation is }3\text{ since the highest order derivative present is } \\ \frac{d^3y}{dx^3}\text{, and the degree is }1\text{ since this derivative appears to the first power.}
\displaystyle \text{The differential equation is non\text{-}linear because the term }\left(\frac{d^2y}{dx^2}\right)^3 \\ \text{ involves a derivative raised to a power greater than }1.

\displaystyle \textbf{Question 13: }~(xy^{2}+x)\,dx+(y-x^{2}y)\,dy=0.
\displaystyle \text{Answer:}
\displaystyle (xy^2+x)\,dx+(y-x^2y)\,dy=0
\displaystyle \Rightarrow x(y^2+1)\,dx=y(x^2-1)\,dy
\displaystyle \Rightarrow \frac{x(y^2+1)}{y(x^2-1)}=\frac{dy}{dx}
\displaystyle \Rightarrow x(y^2+1)\frac{dy}{dx}-y(x^2-1)=0
\displaystyle \Rightarrow (y^2+1)\frac{dy}{dx}-y\left(x-\frac{1}{x}\right)=0
\displaystyle \text{Here, the order of the differential equation is }1\text{ since the highest order derivative present is } \\ \frac{dy}{dx}\text{, and the degree is }1\text{ since it appears to the first power.}
\displaystyle \text{The differential equation is non\text{-}linear because the product }y^2\frac{dy}{dx} \\ \text{ involving the dependent variable and its derivative is present.}

\displaystyle \textbf{Question 14: }~\sqrt{1-y^{2}}\,dx+\sqrt{1-x^{2}}\,dy=0.
\displaystyle \text{Answer:}
\displaystyle \sqrt{1-y^2}\,dx+\sqrt{1-x^2}\,dy=0
\displaystyle \Rightarrow \sqrt{1-y^2}\,dx=-\sqrt{1-x^2}\,dy
\displaystyle \Rightarrow -\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}=\frac{dy}{dx}
\displaystyle \Rightarrow \sqrt{1-x^2}\frac{dy}{dx}+\sqrt{1-y^2}=0
\displaystyle \text{Here, the order of the differential equation is }1\text{ since the highest order derivative present is } \\ \frac{dy}{dx}\text{, and the degree is }1\text{ since it appears to the first power.}
\displaystyle \text{The differential equation is non\text{-}linear because the dependent variable }y \\ \text{ appears inside the non\text{-}linear function }\sqrt{1-y^2}.

\displaystyle \textbf{Question 15: }~\frac{d^{2}y}{dx^{2}}=\left(\frac{dy}{dx}\right)^{2/3}.
\displaystyle \text{Answer:}
\displaystyle \frac{d^2y}{dx^2}=\left(\frac{dy}{dx}\right)^{\frac{2}{3}}
\displaystyle \text{Taking cubes of both sides, we get}
\displaystyle \left(\frac{d^2y}{dx^2}\right)^3=\left(\frac{dy}{dx}\right)^2
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2y}{dx^2}\text{, and the degree is }3\text{ since this derivative appears with highest power }3.
\displaystyle \text{The differential equation is non\text{-}linear because its degree is }3 \\ \text{, which is greater than }1.

\displaystyle \textbf{Question 16: }~2\frac{d^{2}y}{dx^{2}}+3\sqrt{1-\left(\frac{dy}{dx}\right)^{2}}-y=0.
\displaystyle \text{Answer:}
\displaystyle 2\frac{d^2y}{dx^2}+3\sqrt{1-\left(\frac{dy}{dx}\right)^2}-y=0
\displaystyle \Rightarrow 2\frac{d^2y}{dx^2}=-3\sqrt{1-\left(\frac{dy}{dx}\right)^2}+y
\displaystyle \text{Squaring both sides, we get}
\displaystyle 4\left(\frac{d^2y}{dx^2}\right)^2=9\left[1-\left(\frac{dy}{dx}\right)^2\right]-6y\sqrt{1-\left(\frac{dy}{dx}\right)^2}+y^2
\displaystyle \Rightarrow 4\left(\frac{d^2y}{dx^2}\right)^2+9\left(\frac{dy}{dx}\right)^2+y^2-6y\sqrt{1-\left(\frac{dy}{dx}\right)^2}-9=0
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2y}{dx^2}\text{, and the degree is }2\text{ since this derivative appears with highest power }2.
\displaystyle \text{The differential equation is non\text{-}linear because it contains non\text{-}linear terms involving } \\ \text{derivatives and the dependent variable.}

\displaystyle \textbf{Question 17: }~5\frac{d^{2}y}{dx^{2}}=\left\{1+\left(\frac{dy}{dx}\right)^{2}\right\}^{3/2}.
\displaystyle \text{Answer:}
\displaystyle 5\frac{d^2y}{dx^2}=\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}}
\displaystyle \text{Squaring both sides, we get}
\displaystyle 25\left(\frac{d^2y}{dx^2}\right)^2=\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^3
\displaystyle \Rightarrow 25\left(\frac{d^2y}{dx^2}\right)^2=1+3\left(\frac{dy}{dx}\right)^2+3\left(\frac{dy}{dx}\right)^4+\left(\frac{dy}{dx}\right)^6
\displaystyle \Rightarrow 25\left(\frac{d^2y}{dx^2}\right)^2-\left(\frac{dy}{dx}\right)^6-3\left(\frac{dy}{dx}\right)^4-3\left(\frac{dy}{dx}\right)^2-1=0
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2y}{dx^2}\text{, and the degree is }2\text{ since this derivative appears with highest power }2.
\displaystyle \text{The differential equation is non\text{-}linear because its degree is }2\text{, which is greater than }1.

\displaystyle \textbf{Question 18: }~y=x\frac{dy}{dx}+a\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}.
\displaystyle \text{Answer:}
\displaystyle y=x\frac{dy}{dx}+a\sqrt{1+\left(\frac{dy}{dx}\right)^2}
\displaystyle \Rightarrow y-x\frac{dy}{dx}=a\sqrt{1+\left(\frac{dy}{dx}\right)^2}
\displaystyle \text{Squaring both sides, we get}
\displaystyle \left(y-x\frac{dy}{dx}\right)^2=a^2\left[1+\left(\frac{dy}{dx}\right)^2\right]
\displaystyle \Rightarrow y^2-2xy\frac{dy}{dx}+x^2\left(\frac{dy}{dx}\right)^2=a^2+a^2\left(\frac{dy}{dx}\right)^2
\displaystyle \Rightarrow (x^2-a^2)\left(\frac{dy}{dx}\right)^2-2xy\frac{dy}{dx}+(y^2-a^2)=0
\displaystyle \text{Here, the order of the differential equation is }1\text{ since the highest order derivative present is } \\ \frac{dy}{dx}\text{, and the degree is }2\text{ since this derivative appears with highest power }2.
\displaystyle \text{The differential equation is non\text{-}linear because its degree is }2\text{, which is greater than }1.

\displaystyle \textbf{Question 19: }~y=px+\sqrt{a^{2}p^{2}+b^{2}},\ \text{where }p=\frac{dy}{dx}.
\displaystyle \text{Answer:}
\displaystyle y=px+\sqrt{a^2p^2+b^2}
\displaystyle \Rightarrow y-px=\sqrt{a^2p^2+b^2}
\displaystyle \text{Squaring both sides, we get}
\displaystyle (y-px)^2=a^2p^2+b^2
\displaystyle \Rightarrow y^2-2pxy+p^2x^2=a^2p^2+b^2
\displaystyle \Rightarrow (x^2-a^2)p^2-2xy\,p+(y^2-b^2)=0
\displaystyle \Rightarrow (x^2-a^2)\left(\frac{dy}{dx}\right)^2-2xy\frac{dy}{dx}+y^2-b^2=0
\displaystyle \text{Here, the order of the differential equation is }1\text{ since the highest order derivative present is } \\ \frac{dy}{dx}\text{, and the degree is }2\text{ since this derivative appears with highest power }2.
\displaystyle \text{The differential equation is non\text{-}linear because its degree is }2\text{, which is greater than }1.

\displaystyle \textbf{Question 20: }~\frac{dy}{dx}+e^{y}=0.
\displaystyle \text{Answer:}
\displaystyle \frac{d^2y}{dx^2}+3\left(\frac{dy}{dx}\right)^2=x^2\log\left(\frac{d^2y}{dx^2}\right)
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \frac{d^2y}{dx^2}.
\displaystyle \text{Clearly, the right\text{-}hand side of the differential equation cannot be expressed as a polynomial in } \\ \frac{d^2y}{dx^2}\text{ because it involves a logarithmic function.}
\displaystyle \text{Hence, the degree of the differential equation is not defined.}
\displaystyle \text{Therefore, the order of the differential equation is }2\text{ and its degree is not defined.}
\displaystyle \text{The differential equation is non\text{-}linear because the term }\left(\frac{dy}{dx}\right)^2 \\ \text{ involves the derivative raised to a power greater than }1.

\displaystyle \textbf{Question 21: }~\left(\frac{d^{2}y}{dx^{2}}\right)^{2}+\left(\frac{dy}{dx}\right)^{2}=x\sin\left(\frac{d^{2}y}{dx^{2}}\right).
\displaystyle \text{Answer:}
\displaystyle \left(\frac{d^2y}{dx^2}\right)^2+\left(\frac{dy}{dx}\right)^2=x\sin\left(\frac{d^2y}{dx^2}\right)
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order } \\ \text{derivative present is }\frac{d^2y}{dx^2}.
\displaystyle \text{Clearly, the right\text{-}hand side of the differential equation cannot be expressed } \\ \text{as a polynomial in } \frac{d^2y}{dx^2}\text{ because it involves a trigonometric function of the derivative.}
\displaystyle \text{Hence, the degree of the differential equation is not defined.}
\displaystyle \text{Therefore, the order of the differential equation is }2\text{ and its degree is not defined.}
\displaystyle \text{The differential equation is non\text{-}linear because the term }\left(\frac{dy}{dx}\right)^2 \\ \text{ involves the derivative raised to a power greater than }1.

\displaystyle \textbf{Question 22: }~(y'')^{2}+(y')^{3}+\sin y=0.
\displaystyle \text{Answer:}
\displaystyle (y')^2+(y')^3+\sin y=0
\displaystyle \text{Here, the order of the differential equation is }1\text{ since the highest order } \\ \text{derivative present is }y'=\frac{dy}{dx}\text{.}
\displaystyle \text{The degree of the differential equation is }3\text{ since the highest power of the highest } \\ \text{order derivative }y'\text{ is }3.
\displaystyle \text{The differential equation is non\text{-}linear because the derivative appears with } \\ \text{power greater than }1  \text{ and also because of the non\text{-}linear term }\sin y.

\displaystyle \textbf{Question 23: }~\frac{d^{2}y}{dx^{2}}+5x\left(\frac{dy}{dx}\right)-6y=\log x.
\displaystyle \text{Answer:}
\displaystyle \frac{d^2y}{dx^2}+5x\frac{dy}{dx}-6y=\log x
\displaystyle \text{Here, the order of the differential equation is }2\text{ since the highest order derivative present is } \\ \frac{d^2y}{dx^2}\text{, and the degree is }1\text{ since this derivative appears to the first power.}
\displaystyle \text{The given differential equation is linear.}

\displaystyle \textbf{Question 24: }~\frac{d^{3}y}{dx^{3}}+\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}+y\sin y=0.
\displaystyle \text{Answer:}
\displaystyle \frac{d^3y}{dx^3}+\frac{d^2y}{dx^2}+\frac{dy}{dx}+y\sin y=0
\displaystyle \text{Here, the order of the differential equation is }3\text{ since the highest order derivative present is } \\ \frac{d^3y}{dx^3}\text{, and the degree is }1\text{ since this derivative appears to the first power.}
\displaystyle \text{The differential equation is non\text{-}linear because the term }y\sin y \\ \text{ is a non\text{-}linear function of the dependent variable }y.

\displaystyle \textbf{Question 25: }~\frac{d^{2}y}{dx^{2}}+3\left(\frac{dy}{dx}\right)^{2}=x^{2}\log\left(\frac{d^{2}y}{dx^{2}}\right).
\displaystyle \text{Answer:}
\displaystyle \frac{dy}{dx}+e^{y}=0
\displaystyle \text{Here, the order of the differential equation is }1\text{ since the highest order } \\ \text{derivative present is } \frac{dy}{dx}\text{, and the degree is }1\text{ since this derivative appears to the first power.}
\displaystyle \text{The differential equation is non\text{-}linear because the dependent } \\ \text{variable }y \text{ appears in the exponential form }e^{y}\text{, which is a non\text{-}linear function of }y.

\displaystyle \textbf{Question 26: }~\left(\frac{dy}{dx}\right)^{3}-4\left(\frac{dy}{dx}\right)^{2}+7y=\sin x.
\displaystyle \text{Answer:}
\displaystyle \left(\frac{dy}{dx}\right)^3-4\left(\frac{dy}{dx}\right)^2+7y=\sin x
\displaystyle \text{Here, the order of the differential equation is }1\text{ since the highest order } \\ \text{derivative present is }\frac{dy}{dx}\text{.}
\displaystyle \text{The degree of the differential equation is }3\text{ since the highest power of the } \\ \text{highest order derivative } \frac{dy}{dx}\text{ is }3.
\displaystyle \text{The differential equation is non\text{-}linear because the derivative } \\ \frac{dy}{dx}\text{ appears with power greater than }1.

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