We have shown that in the case of complex conjugate roots,r =a±ib,b̸=0, of the indicial equation, the general solution to the Cauchy-Euler equation y′′+a1y′+a2y =0, x > 0 is y(x)=xa[c1 cos(b ln x)+c2 sin(b ln x)]. .........................(8.8.26) Q. Show that (8.8.26) can be written in the form y(x)= Axa cos(b ln x −φ) for appropriate constants A and φ.
We have shown that in the case of complex conjugate roots,r =a±ib,b̸=0, of the indicial equation, the general solution to the Cauchy-Euler equation y′′+a1y′+a2y =0, x > 0 is y(x)=xa[c1 cos(b ln x)+c2 sin(b ln x)]. .........................(8.8.26) Q. Show that (8.8.26) can be written in the form y(x)= Axa cos(b ln x −φ) for appropriate constants A and φ.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.2: Complex Numbers And Quaternions
Problem 45E
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We have shown that in the case of complex conjugate roots,r =a±ib,b̸=0, of the indicial equation, the general solution to the Cauchy-Euler equation
y′′+a1y′+a2y =0, x > 0 is
y(x)=xa[c1 cos(b ln x)+c2 sin(b ln x)]. .........................(8.8.26)
Q. Show that (8.8.26) can be written in the form y(x)= Axa cos(b ln x −φ) for appropriate constants A and φ.
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