Solve CauchyEuler equation Solve CauchyEuler equation 4t2 y

Solve Cauchy-Euler equation

Solve Cauchy-Euler equation 4t^2 y\" + 4ty\' - y = 12/t

Solution

Given Cauchy-Euler equation is 4t²y\'\'+4ty\'-y=12/t

We can rewrite this Cauchy-Euler equation as

t²y\'\'+ty\'-y/4=3/t   ...(1)

First we will consider the homogeneous equation

t²y\'\'+ty\'-y/4=0   ...(2)

Let y=t^r

y\'=rt^r-1

ty\'=rt^r

y\'\'=r(r-1)t^r-2

t²y\'\'=r(r-1)t^r

Now putting the value of t²y\'\', ty\' and yin the equation (2), we get

r(r-1)t^r+rt^r-t^r/4=0

t^r(r²-r+r-1/4)=0

t^r(r²-1/4)=0

t²(r-1/2)(r+1/2)=0

Hence r=1/2, -1/2

The general solution for equation (2) is

c₁t^1/2+c₂t^-1/2

Now for the non-homogeneous equation (1)

We guess a perticular solution

y_p(t)=A/t

Then y\'_p(t)= -A/t²

y\'\'_p(t)=2A/t³

Hence putting y_p, y\'_p and y\'\'_p back in equation (1), we get

t²×2A/t³+ t× (-A/t²)-A/4t=3/t

2A/t-A/t-A/4t=3/t

1/t(2A-A-A/4)=3/t

3A/4=3

A=4

So y_p(t)=4/t

Hence the general solution for equation (1) is

y(t)= c₁t^1/2+c₂t^-1/2+4/t

So the solution for the given Cauchy-Euler equation is

y(t)= c₁t^1/2+c₂t^-1/2+4/t