Q.

Determine the order and the degree of the following differential equations:

➤

1. \(2x\frac{dy}{dx}+3y=0\)
   Solution:
   Order: 1
   Degree: 1
2. \((\frac{dy}{dx})^2=4x\)
   Solution:
   Order: 1
   Degree: 2
3. \(\frac{d^2 y}{2x^2}+2x=0\)
   Solution:
   Order: 2
   Degree: 1
4. \(x(\frac{d^3 y}{dx^3})^2+(dy/dx)^4 +5y =0\)
   Solution: 
   Order: 3
   Degree: 2
5. \((\frac{d^3 y}{dx^3})^3 +(\frac{dy}{dx})^5+8x =0\)
   Solution:
   Order: 3
   Degree: 3

Q.

Form the differential equations of the following curves:

➤

1. \( y = mx \)
   Solution:
   \begin{align*}
          \frac{dy}{dx} &= m \\
          \frac{dy}{dx} &= \frac{y}{x}
      \end{align*}
2. \( x^2 + y^2 = a^2 \)
   Solution:
   \begin{align*}
          2x + 2y \frac{dy}{dx} &= 0 \\
          \frac{dy}{dx} &= -\frac{x}{y}
      \end{align*}
3. \( y = 2x + ax^2 \)
   Solution:
    \begin{align*}
          \frac{dy}{dx} &= 2 + 2ax \\
          \frac{dy}{dx} &= 2 + 2x\frac{y-2x}{x^2} \\
          \frac{dy}{dx} &= 2 + \frac{2y}{x} - 4 \\
          \frac{dy}{dx} &= \frac{2y}{x} - 2 \\
          x\frac{dy}{dx} - 2y + 2x &= 0 \\
      \end{align*}
4. \( y = ae^x + be^{-x} \)
   Solution:
   \begin{align*}
          \frac{dy}{dx} &= ae^x - be^{-x} \\
          \frac{d^2y}{dx^2} &= ae^x + be^{-x} \\
          \frac{d^2y}{dx^2} &= y \\
          \frac{d^2y}{dx^2} - y &= 0
      \end{align*}

Q.

➤

\begin{document}

\begin{itemize}
   \item[a)] \( y = mx \)
   \begin{align*}
       \frac{dy}{dx} &= m \\
       \frac{dy}{dx} &= \frac{y}{x}
   \end{align*}

   \item[b)] \( x^2 + y^2 = a^2 \)
   \begin{align*}
       2x + 2y \frac{dy}{dx} &= 0 \\
       \frac{dy}{dx} &= -\frac{x}{y}
   \end{align*}

   \item[c)] \( y = 2x + ax^2 \)
   \begin{align*}
       \frac{dy}{dx} &= 2 + 2ax \\
       \frac{dy}{dx} &= 2 + 2x\frac{y-2x}{x^2} \\
       \frac{dy}{dx} &= 2 + \frac{2y}{x} - 4 \\
       \frac{dy}{dx} &= \frac{2y}{x} - 2 \\
       x\frac{dy}{dx} - 2y + 2x &= 0 \\
   \end{align*}

   \item[d)] \( y = ae^x + be^{-x} \)
   \begin{align*}
       \frac{dy}{dx} &= ae^x - be^{-x} \\
       \frac{d^2y}{dx^2} &= ae^x + be^{-x} \\
       \frac{d^2y}{dx^2} &= y \\
       \frac{d^2y}{dx^2} - y &= 0
   \end{align*}
\end{itemize}

\end{document}