Exercises

(1)
In attempting to use the Reduction of Order formula
\(y_1 z\,^{\prime } +(2y_1\,^{\prime }+py_1)z=f\) to solve \(x^2 y\,^{\prime \prime } +axy\,^{\prime }-ay=2x\), where \(a\) is a nonzero constant, given that \(y_1=x\) satisfies \(x^2 y\,^{\prime \prime } +a xy\,^{\prime }-ay=0\), what should \(p,f\) be?
(a)
\(p=x\), \(f=2x\).
(b)
\(p=ax\), \(f=2x\).
(c)
\(p=-ax^{-1}\), \(f=2x\).
(d)
\(p=ax^{-1}\), \(f=2x^{-1}\).
(e)
\(p=ax^{-2}\), \(f=2x^{-1}\)
(f)
\(p=x^2\), \(f=2x^{-1}\)
(2)
Suppose Reduction of Order is used to attempt to solve
\(xy^{\,\prime \prime }-(2x+1)y^{\,\prime }+(x+1)y=-e^x\), given that \(y_1=e^x\) is a solution to
\(xy^{\,\prime \prime }-(2x+1)y^{\,\prime }+(x+1)y=0\). What first order equation (in standard form) would \(z=u^{\,\prime }\) satisfy, where \(u=\dfrac {y}{e^x}\)?
(a)
\(z^{\,\prime }+\dfrac {z}{x} = -1\)
(b)
\(z^{\,\prime }-\dfrac {z}{x} = -\dfrac {1}{x}\)
(c)
\(z^{\,\prime } +(2-2xe^{-x}-e^{-x})z=-1\)
(d)
\(z^{\,\prime } -2e^x z =-\dfrac {1}{x}\)
(e)
\(z^{\,\prime } +2e^x z =-\dfrac {1}{x}\)
(f)
\(z^{\,\prime } - 2xe^x z =-e^x\)
(3)
Suppose \(y(x)\) is the solution to \(xy^{\,\prime \prime }-(2x+1)y^{\,\prime }+(x+1)y=-e^x\) involving the least number of terms. Determine \(y(1)\). Round to the nearest hundredth.
(4)
Consider the second order linear homogeneous differential equation
\(xy\,^{\prime \prime }-(4x+1)y\,^{\prime }+(4x+2)y=0\) on \((0,\infty )\). Find a solution to this equation of the form \(y_1=e^{kx}\) for some constant \(k\). Then use Reduction of Order to determine the solution \(y_2\) with the fewest terms satisfying \(y_2\neq cy_1\) for any constant \(c\) and \(y_2(1)=e^2\). Determine \(y_2(2)\). Round to the nearest hundredth.
(5)
Suppose \(y(t)\) is the solution to \(y^{\,\prime \prime }-3y^{\,\prime }+2y=-\dfrac {1}{1+e^{-t}}\) involving the least number of terms. Find \(y(1)\). Round to the nearest hundredth.

Hints: use \(y_1=e^{2t}\) (rather than \(e^t\)); you may find
\(\displaystyle \int \ln (x)\,\mathrm {d}x=x\ln (x)-x+C\) useful.