MATH SOLVE

3 months ago

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

Accepted Solution

A:

Answer:yesStep-by-step explanation:We are given that a Cauchy Euler's equation [tex]t^2y''-ty'+y=0[/tex] where t is not equal to zeroWe are given that two solutions of given Cauchy Euler's equation are t,t ln tWe have to find the solutions are independent or dependent.To find the solutions are independent or dependent we use wronskain [tex]w(x)=\begin{vmatrix}y_1&y_2\\y'_1&y'_2\end{vmatrix}[/tex]If wrosnkian is not equal to zero then solutions are dependent and if wronskian is zero then the set of solution is independent.Let [tex]y_1=t,y_2=t ln t[/tex][tex]y'_1=1,y'_2=lnt+1[/tex][tex]w(x)=\begin{vmatrix}t&t lnt\\1&lnt+1\end{vmatrix}[/tex][tex]w(x)=t(lnt+1)-tlnt=tlnt+t-tlnt=t [/tex] 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.