Exercises

(1)
Let \(p(x),q(x)\) be continuous functions on \((0,\infty )\). Suppose \(y_1\) and \(y_2\) form a fundamental set of solutions satisfying \(\displaystyle y\,^{\prime \prime }+p(x)y\,^{\prime }+q(x)y=0\) on \((0,\infty )\). Which of the following CANNOT be the Wronskian \(W(x)\) of \(y_1\) and \(y_2\)? Select all that apply.
(a)
\(W(x)=x\)
(b)
\(W(x)=x-1\)
(c)
\(W(x)=x+1\)
(d)
\(W(x)=\dfrac {1}{x}\)
(e)
\(W(x)=4x^2-1\)
(f)
\(W(x)=1\)
(g)
\(W(x)=e^x-1\)
(h)
\(W(x)=e^x-2\)
(2)
Let \(W(x)\) be the Wronskian of \(y_1=\ln {\left (\sqrt {x^3}\right )}\) and \(\displaystyle y_2=2\ln {\left (\frac {1}{x^2}\right )}\). Determine \(W(2026)\).
(3)
Suppose \(y_1\) and \(y_2\) are two solutions to \(\displaystyle (2x+1) y''+2 y'-y=0\) on the interval \((-1/2,\infty )\) and their Wronskian \(W(x)\) satisfies \(W(0)=1\). Determine \(W(2026)\). Round to the nearest ten-thousandth.
(4)
Which is the Wronskian of some pair of linearly independent solutions to \(x^2y^{\,\prime \prime }-x(x+2)y^{\,\prime }+(x+2)y=0\) for \(x>0\)?
(a)
\(0\)
(b)
\(x+\ln (x^2)\)
(c)
\(x\ln (x^2)\)
(d)
\(x+e^x\)
(e)
\(xe^x\)
(f)
\(x^2+e^x\)
(g)
\(x^2e^x\)
(h)
None of these
(5)
Given two linearly independent solutions \(y_1,y_2\) to a second order linear homogeneous differential equation, for any solution \(y\) to that same equation, the functions \(y_1,y_2,y\) must form a linearly dependent set, which implies (via linear algebra) that
\[\displaystyle \mathrm {det}\left (\begin{array}{ccc} y_1 & y_2 & y\\ y_1' & y_2' & y'\\ y_1'' & y_2'' & y''\end{array}\right ) = y_1\,\mathrm {det}\left (\begin{array}{cc} y_2' & y'\\ y_2'' & y'' \end{array}\right )-y_2\,\mathrm {det}\left (\begin{array}{cc} y_1' & y'\\ y_1'' & y'' \end{array}\right ) + y\,\mathrm {det}\left (\begin{array}{cc} y_1' & y_2'\\ y_1'' & y_2'' \end{array}\right )=0.\]
Use this to construct a second order linear homogeneous differential equation for which \(y_1=x\) and \(y_2=e^x\) are linearly independent solutions. Let \(p(x)\) be the function in front of \(y^{\,\prime }\) when the equation is put into standard form. Determine \(p(-1)\).

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