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Chapter 4: Q10E (page 199)

In Problems 9 through 14, find a general solution to the given Cauchy鈥揈uler equation for t>0.t2d2ydt2+2tdydt-6y=0

Short Answer

Expert verified

The general equation isy=c1t-3+c2t2.

Step by step solution

01

Find the auxiliary equation. 

Given differential equation t2d2ydt2+2tdydt-6y=0鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�鈥�(1)

Assume y=trthen we have;

y'=rtr-1y''=r(r-1)tr-2

Substitute all values in equation (1), and we get:

t2r(r-1)tr-2+2trtr-1-6tr=0(r(r-1)+2r-6)tr=0r2+r-6=0

02

Determine the general equation. 

The roots of the equation are:

r2+3r-2r-6=0(r+3)(r-2)=0r=-3,2

Thus, the general solution is y=c1t-3+c2t2.

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