Question

Find the Wronskians of the following sets of functions and statewhat that Wronskian implies about
linear independence or dependence of the set.
(5) 2x + 11; 22x + 121 on (-∞,∞)
(6) sin(x); sin(2x) on (-∞,∞)
(7) x² + x - 2; x² + x; x² + x - 6on (-∞,∞)
(8) x² + x - 2; x² + x + 1; x² + x- 5 on (-∞,∞)

Answer 1

we know that the functions \(f(x), g(x)\) and \(h(x)\) are linearlyindependent if there wronskian is not equal to zero. wronskian isgiven by the determinant \(W=\left|\begin{array}{lll}f & g & h \\ f^{\prime} & g^{\prime} & h^{\prime} \\ f^{\prime \prime} & g^{\prime \prime} & h^{\prime \prime}\end{array}\right|\)

5) let \(f=2 x+11, g(x)=22 x+121\)

\(\Rightarrow_{f^{\prime}}=2, g^{\prime}=22\)

\(W=\left|\begin{array}{cc}2 x+11 & 22 x+121 \\ 2 & 22\end{array}\right|\)

\(w=22(2 x+11)-2(22 x+121)\)

\(\mathrm{W}=0\)

hence the functions are linearly dependent. 6) let \(f(x)=\sin (x), g(x)=\sin (2 x)\)

\(\Rightarrow f^{\prime}=\cos (x), g^{\prime}(x)=2 \cos 2 x\)

\(W=\left|\begin{array}{cc}f & g \\ f^{\prime} & g^{\prime}\end{array}\right|=\left|\begin{array}{cc}\sin x & \sin 2 x \\ \cos x & 2 \cos 2 x\end{array}\right|=2 \sin (x) \cos (2 x)-2 \sin 2(x) \cos (x) \neq 0\)

hence the functions are linearly independent (7) let \(f(x)=x^{2}+x-2 g(x)=x^{2}+x+1 h(x)=x^{2}+x-5\)

\(\Rightarrow f^{\prime}=2 x+1, g^{\prime}=2 x+1 h^{\prime}=2 x+1\)

\(\Rightarrow \mathrm{f"}=2, \mathrm{~g}^{\prime \prime}=2, \mathrm{~h}^{\prime \prime}=0\)

\(W=\left|\begin{array}{ccc}f & g & h \\ f^{\prime} & g^{\prime} & h^{\prime} \\ f^{\prime \prime} & g^{\prime \prime} & h^{\prime \prime}\end{array}\right|\)

\(W=\left|\begin{array}{ccc}x^{2}+x & x^{2}+x+1 & x^{2}+x-5 \\ 2 x+1 & 2 x+1 & 2 x+1 \\ 2 & 2 & 2\end{array}\right|\)

\(=\left|\begin{array}{ccc}x^{2}+x & x^{2}+x+1 & x^{2}+x-5 \\ 2 x+1 & 0 & 0 \\ 2 & 0 & 0\end{array}\right|=0\)

hence the functions are linearly dependent