Question

convert the equation xy"-3y'+x^-1y = x^2 as a linear equation with constant coefficient

Answer 1

Answer:

We are given:

x2y′′+y=0,  x>0(1)(1)x2y″+y=0,  x>0

This is a Euler-Cauchy type DEQ.

We can let y=xmy=xm, so we have: y′(x)=mxm−1,  y′′(x)=m(m−1)xm−2y′(x)=mxm−1,  y″(x)=m(m−1)xm−2.

Substituting this back into (1)(1), yields:

x2y′′+y=x2(m(m−1)xm−2)+xm=xm(m2−m+1)=0x2y″+y=x2(m(m−1)xm−2)+xm=xm(m2−m+1)=0.

So, we have a characteristic equation:

m2−m+1→m1,2=

Step-by-step explanation:

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