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Cauchy Euler Equation
Question 1
For the Cauchy-Euler equation x2y′′+ axy′+by = 0x2 , the substitution y = xm leads to:
A quadratic equation in x
A quadratic equation in m
C) A linear equation in mmm
A cubic equation in m
Question 2
Which of the following equations is of Cauchy-Euler type?
x2y′′−3xy′+4y = 0x2
y′′+ xy′+ y = 0
y′′+ exy = 0
y′′+ y = 0
Question 3
Which of the following statements is TRUE for Cauchy-Euler equations?
They always have constant coefficients
The substitution y=emxy = e^{mx}y=emx is used to solve them
They can be transformed into equations with constant coefficients by the substitution x =et
They cannot have repeated roots in their characteristic equation
Question 4
If a Cauchy-Euler equation has a repeated root m = r, the general solution is:
y = C1xr + C2xr
y = C1xr + C2x2r
y = C1xr + C2xr ln x
y = C1erx + C2xr
Question 5
Which substitution is used to transform a Cauchy-Euler equation into a constant coefficient differential equation?
y = xm
x = et
y=emx
x = ln t
Question 6
Which of the following is not a Cauchy-Euler equation?
x2y′′+xy′− 4y = 0
x3y′′′+ 2x2y′′−xy′+y = 0
y′′+ 3y′+ 2y = 0
x2y′′−5xy′+6y = 0
There are 6 questions to complete.