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Q1 Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align} Ans: \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0 \end{align}
\begin{align} \Rightarrow y^{m\;'}+\;\sin(y^m)\;=0 \end{align}
The highest order derivative present in the differential equation is ym '. Therefore, its order is four.
The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.
Q2 Determine order and degree(if defined) of differential equation y' + 5y = 0 Ans: The given differential equation is:
y' + 5y = 0
The highest order derivative present in the differential equation isy'. Therefore, its order is one.
It is a polynomial equation in y'. The highest power raised to y' is 1. Hence, its degree is one.
Q3 Determine order and degree(if defined) of differential equation \begin{align}\left(\frac{ds}{dt}\right)^4\;+\;3s\frac{d^2s}{dt^2}\;=\;0\end{align} Ans: \begin{align}\left(\frac{ds}{dt}\right)^4\;+\;3s\frac{d^2s}{dt^2}\;=\;0\end{align}
The highest order derivative present in the given differential equation is\begin{align}\frac{d^2s}{dt^2}.\end{align}
Therefore, its order is two. It is a polynomial equation in
\begin{align}\frac{d^2s}{dt^2} and \frac{ds}{dt}.\end{align}
The power raised to is 1. \begin{align} \frac{d^2s}{dt^2} \end{align}
Q4 Determine order and degree(if defined) of differential equation \begin{align}\left(\frac{d^2y}{dx^2}\right)^2\;+\;cos\left(\frac{dy}{dx}\right)\;=\;0\end{align} Ans: \begin{align}\left(\frac{d^2y}{dx^2}\right)^2\;+\;cos\left(\frac{dy}{dx}\right)\;=\;0\end{align}
The highest order derivative present in the given differential equation is \begin{align}\frac{d^2y}{dx^2}.\end{align}
Therefore, its order is 2. The given differential equation is not a polynomial equation in its derivatives.
Hence, its degree is not defined.
Q5 Determine order and degree(if defined) of differential equation \begin{align}\frac{d^2y}{dx^2}=\cos3x + sin3x\end{align} Ans: \begin{align}\frac{d^2y}{dx^2}=\cos3x + sin3x\end{align}
\begin{align}\Rightarrow\frac{d^2y}{dx^2} - \cos3x - sin3x = 0\end{align}
The highest order derivative present in the differential equation is\begin{align}\frac{d^2y}{dx^2}.\end{align}
Therefore, its order is two.It is a polynomial equation in \begin{align}\frac{d^2y}{dx^2}\end{align}
and the power raised to is 1.
\begin{align}\frac{d^2y}{dx^2}\end{align}
Hence, its degree is one.
Q6 Determine order and degree(if defined) of differential equation (ym)2 + (yn)3 + (y')4 + y5 =0 Ans: (ym)2 + (yn)3 + (y')4 + y5 =0
The highest order derivative present in the differential equation isym. Therefore, its order is three.
The given differential equation is a polynomial equation in ym , yn , y'.
The highest power raised to ym is 2. Hence, its degree is 2.
Q7 Determine order and degree(if defined) of differential equation ym + 2yn + y' =0 Ans: The highest order derivative present in the differential equation is ym. Therefore, its order is three.
It is a polynomial equation in ym , yn and y' . The highest power raised to ym is 1. Hence, its degree is 1.
Q8 Determine order and degree(if defined) of differential y' + y =ex Ans: y' + y =ex
y' + y - ex =0
The highest order derivative present in the differential equation is y'. Therefore, its order is one.
The given differential equation is a polynomial equation in y' and the highest power raised to y' is one. Hence, its degree is one.
Q9 Determine order and degree(if defined) of differential equation yn + (y')2 + 2y =0 Ans: yn + (y')2 + 2y =0
The highest order derivative present in the differential equation is yn. Therefore, its order is two.
The given differential equation is a polynomial equation in yn and y' and the highest power raised to yn is one.
Hence, its degree is one.
Q10 Determine order and degree(if defined) of differential equation yn + 2y' + siny = 0 Ans: yn + 2y' + siny = 0
The highest order derivative present in the differential equation is yn. Therefore, its order is two.
This is a polynomial equation in yn and y' and the highest power raised to yn is one. Hence, its degree is one.
Q11 The degree of the differential equation \begin{align}\left(\frac{d^2y}{dx^2}\right)^3\;+ \left(\frac{dy}{dx}\right)^2+\;sin\left(\frac{dy}{dx}\right)\;+ 1=\;0\end{align} is (A) 3 (B) 2 (C) 1 (D) not defined Ans: \begin{align}\left(\frac{d^2y}{dx^2}\right)^3\;+ \left(\frac{dy}{dx}\right)^2+\;sin\left(\frac{dy}{dx}\right)\;+ 1=\;0\end{align}
The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined.
Hence, the correct answer is D.
Q12 The order of the differential equation \begin{align}2x^2\frac{d^2y}{dx^2}\;- \;3\frac{dy}{dx}\;+ y=\;0\end{align} is (A) 2 (B) 1 (C) 0 (D) not defined Ans: \begin{align}2x^2\frac{d^2y}{dx^2}\;- \;3\frac{dy}{dx}\;+ y=\;0\end{align}
The highest order derivative present in the given differential equation is
\begin{align}\frac{d^2y}{dx^2}\end{align}
Therefore, its order is two.
Hence, the correct answer is A.