Question

Does the set {t, t Int} form a fundamental set of solutions for t^2y" -- ty' +y = 0?

Answer 1

yes

Explanation:

We are given that a Cauchy Euler's equation

`t^2y''-ty'+y=0` where t is not equal to zero

We are given that two solutions of given Cauchy Euler's equation are t,t ln t

We have to find the solutions are independent or dependent.

To find the solutions are independent or dependent we use wronskain

`w(x)=\begin{vmatrix}y_1&y_2\\y'_1&y'_2\end{vmatrix}`

If wrosnkian is not equal to zero then solutions are dependent and if wronskian is zero then the set of solution is independent.

Let
`y_1=t,y_2=t ln t`

`y'_1=1,y'_2=lnt+1`

`w(x)=\begin{vmatrix}t&t lnt\\1&lnt+1\end{vmatrix}`

`w(x)=t(lnt+1)-tlnt=tlnt+t-tlnt=t` where t is not equal to zero.

Hence,the wronskian is not equal to zero .Therefore, the set of solutions is independent.

Hence, the set {t , tln t} form a fundamental set of solutions for given equation.