Exercises

(1)
Which of the following cannot be a solution to a homogeneous Euler equation (for \(x>0\))? Select all that apply.
(a)
\(\displaystyle y=\frac {x^2-1}{x}\)
(b)
\(\displaystyle y=\frac {1}{x}+\frac {\ln (x)}{x^2}\)
(c)
\(y=(1+\ln (x^{-2}))\sqrt {x^3}\)
(d)
\(y=x\cos {\left (7\ln {(x)}\right )}\)
(e)
\(y=x^2\sin {\left (\ln {(x)}\right )}-x^2\cos {\left (\ln {(x)}\right )}\)
(f)
\(y=\sin {\left (\ln {(x)}\right )}+x\cos {\left (\ln {(x)}\right )}\)
(2)
Suppose \(y(x)\) on \((0,\infty )\) satisfies \(x^2 y^{\,\prime \prime } +3xy^{\,\prime }+5y=-2\), \(y(1)=0\), \(y'(1)=0\). Find \(\displaystyle \lim _{x\rightarrow \infty }y(x)\). If the limit is finite, enter the value of the limit to the nearest tenth. If the limit is \(-\infty \) enter \(-\pi \) to the nearest thousandth. If the limit is \(\infty \) enter \(\pi \) to the nearest thousandth.
(3)
Suppose \(y(x)\) on \((0,\infty )\) satisfies \(4x^2 y''+8xy'+y=0\), \(y(1)=0\), \(y'(1)=1\). Determine \(y(2026)\). Round to the nearest ten-thousandth.
(4)
Suppose \(y(x)\) satisfies \(y\,^{\prime \prime }=\dfrac {20(y+1)}{x^2}\) on \((0,\infty )\) and the graph of \(y(x)\) touches, but does not cross the \(x\)-axis at \((1,0)\). Determine \(y(2)\). Round the nearest thousandth.
(5)
Let \(k\) be a real number. Suppose that any solution to
\(\displaystyle y^{\,\prime \prime } = -\frac {ky}{x^2}\) on \((0,\infty )\) has infinitely many zeros. Which of the following real numbers could be \(k\)? Select all that apply.
(a)
\(-2\)
(b)
\(-1\)
(c)
\(-0.5\)
(d)
\(-0.25\)
(e)
\(0\)
(f)
\(0.25\)
(g)
\(0.5\)
(h)
\(1\)
(i)
\(2\)