Exercises

(1)
Let \(y(t)\) be the particular solution to \(y^{\,\prime \prime }+9ty=t^2\) that involves the least number of non-zero terms. Determine \(y(2026)\). Round to the nearest tenth.
(2)
Let \(I\) be an open interval and assume \(p,q,f\) are continuous functions on \(I\) where \(f\neq 0\). Suppose \(y_{p_1}\) and \(y_{p_2}\) are distinct particular solutions to \(y^{\,\prime \prime }+py^{\,\prime }+qy=f\) on \(I\). Which of the following would be a solution to the complementary equation \(y^{\,\prime \prime }+py^{\,\prime }+qy=0\)?
(a)
\(y_{p_1}+y_{p_2}\)
(b)
\(y_{p_1}-y_{p_2}\)
(c)
\(2y_{p_1}+y_{p_2}\)
(d)
\(y_{p_1}-2y_{p_2}\)
(e)
\(2y_{p_1}-y_{p_2}\)
(3)
Let \(p\) be a continuous function on an open interval \(I\). Furthermore, suppose \(y_1,y_2\) form a fundamental set of solutions to \(y^{\,\prime \prime }+py^{\,\prime }-y=0\) on \(I\). Determine which below are solutions to \(y^{\,\prime \prime }+py^{\,\prime }-y=-1\) on \(I\). Select all that apply.
(a)
\(-1\)
(b)
\(0\)
(c)
\(1\)
(d)
\(1-y_2\)
(e)
\(y_2-1\)
(f)
\(y_1+y_2-1\)
(g)
\(1-y_1-y_2\)
(h)
\(1-\displaystyle \frac {1}{2}y_1\)
(4)
Suppose \(y(t)\) satisfies the IVP \(y^{\,\prime \prime \prime }+y^{\,\prime }=2t\), \(y(0)=-2\), \(y^{\,\prime }(0)=1\), \(y^{\,\prime \prime }(0)=3\) on \((-\infty ,\infty )\). Find \(y(1)\). Round to the nearest tenth.
(5)
Let \(p,q,f\) be continuous functions on an open interval \((0,\infty )\). Suppose that three solutions to \(y^{\,\prime \prime }+p(x)y^{\,\prime }+q(x)y=f(x)\) on \((0,\infty )\) are
\(y_1(x)=\dfrac {\ln {(x)}}{x}+\ln {(x)}\), \(y_2(x)=\dfrac {\ln {(x)}}{2x}+\ln {(x)}\), and
\(y_3(x)=\dfrac {1-\ln {(x)}}{x}+\ln {(x)}\). Suppose \(y(x)\) is the solution to the IVP \(y^{\,\prime \prime }+p(x)y^{\,\prime }+q(x)y=f(x)\), \(y(1)=0\), \(y^{\,\prime }(1)=0\) on \((0,\infty )\). Determine \(y(2026)\). Round to the nearest hundredth.