First Order but higher degree differential equation:
A differential equation involving first order but degree higher than one is called first order but higher degree differential equation.
Example:
\( P^n+A_1 P^{n-1}+A_2 P^{n-2}+\ldots +A_{n-a} P+ A_n\)
is a first-order and n-th degree differential equation, where \(p=\frac{d y}{d x}\) and \(A_1, A_2, \ldots , A_n\)
are functions of x,y
Convenient of solution and various types of form, The equation is solved by five methods,
- Solvable for p
- Solvable for y
- solvable for x
- Clairauit's equation
- Lagrange's equation
Solvable for p:
If the equation
\(P^n+A_1 P^{n-1}+A_2 P^{n-2}+\ldots +A_{n-a} P+ A_n=0\)
can be expressed into n-factors of the first order and first degree
\((p-\alpha_1) (p-\alpha_2)(p-\alpha_3)\ldots (p-\alpha_n)=0 \)
then equating to zero each factor, n-equation of the first order and first degree are obtained, Then the solutions are obtained by integration.
Question: Solve, \(p^2+2xp-3x^2=0 \)
Solution: Given equation is,
\begin{align*}
& p^2+2xp-3x^2=0 \tag{1}\label{eq:1} \\
or, ~& p^2+3xp-xp-3x^2=0\\
or, ~& p(p+3x)-x(p+3x)=0 \\
or, ~& (p-x)(p + 3x)= 0\\
~& \\
\therefore & ~ p - x = 0 \tag{2}\label{eq:2} \\
& ~ p + 3x = 0 \tag{3}\label{eq:3} \\
\end{align*}
From (2) we get,
\begin{align*}
& \frac{dy}{dx} - x = 0\\
& dy - x dx = 0
\end{align*}
Integrating this we get,
\begin{align*}
& y-\frac{x ^ 2 }{2}- \frac{c}{2}= 0 \\
or, ~ & 2y-x ^2 -c = 0
\end{align*}
From (3) we get,
\begin{align*}
& \frac{dy}{dx} +3x =0 \\
or,~ & dy + 3x~dx = 0
\end{align*}
Integrating this we get,
\begin{align*}
& y + \frac{3x2 ^2}{2} - \frac{c}{2}= 0\\
& 2y + 3x^2 - c = 0
\end{align*}
General solution of (1) is,
\begin{align*}
(2y - x^2 - c )(2y + 3x^2 - c ) = 0
\end{align*}
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