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Chapter 8: Q1E (page 450)
In Problems 1-10, use a substitution y=xrto find the general solution to the given equation for x>0.
x2y"(x)+6xy'(x)+6y(x)=0
The general solution to the given equation x2y"(x)+6xy'(x)+6y(x)=0 is y=c1x-2+c2x-3.
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Question : find the power series expansion
for given the expansions for f(x) and g(x).
In Problems 11 and 12, use a substitution of the form to find a general solution to the given equation for x>c.
4(x+2)2y"+5y=0
In Problems 15-17, solve the given initial value problem t2x"-12x=0. x(1)=3 and x'(1)=5.
In Problems \(5 - 14\) solve the given linear system.
\({\bf{X'}} = \left( {\begin{array}{*{20}{c}}{{\rm{ 0 2 1}}}\\{1{\rm{ }}1{\rm{ }} - 2}\\{2{\rm{ }}2{\rm{ }} - 1}\end{array}} \right){\bf{X}}\)
Use the change of variables \(s = \frac{2}{\alpha }\sqrt {\frac{k}{m}} {e^{ - \alpha t/2}}\)to show that the differential equation of the aging spring \(mx'' + k{e^{ - \alpha t}}x = 0\),\(\alpha > 0\)becomes\({s^2}\frac{{{d^2}x}}{{d{s^2}}} + s\frac{{dx}}{{ds}} + {s^2}x = 0\).
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