Lecture 40: Euler’s Equation

40.1 Euler's Equation

40.1.1 Basic Concepts

#####Definition: Eulersequation

description:

1. Format: $$x^ny^{(n)}+a_1x^{n-1}y^{(n-1)}+\cdots+a_{n-1}xy^{\prime}+a_ny=f(x)$$

Explanation

• Concept:
• Euler's equation, which is also a linear equation but has variable coefficients;
• But it is a special type of variable coefficient: a veryregularlinear coefficient equation;
• Philosophy:
• Transform the variable-coefficient equation of this pattern into a constant coefficient equation;
• Method:
• Let $x=e^t$ to get $x^{k}y^{(k)}=D(D-1)\cdots(D-k+1)y$
• Among them, $D=\frac{d}{dt}$, $Dy=\frac{dy}{dt}$, $D^{2}=\frac{d^{2}}{dt^{2}}\quad\quad D^{2}y=\frac{d^{2}y}{dt^{2}}$

40.1.2 Example Problems

Example Problem: Find a general solution to Euler's equation $x^{2}\frac{d^{2}y}{dx^{2}}+4x\frac{dy}{dx}+2y=0\quad(x>0)$;

• Analysis
• Analysis
• $x=e^{t}\quad D(p-1)y+4Dy+2y=0$
• Get: $(y^{2}-1)y+4y+2y=0$
• De: $\frac{d^{2}y}{dt^{2}}+3\frac{dy}{dt}+2y=0$
• Question Type: #