Exercises

(1)
Given that \(y(t)\) satisfies \(y^{\,\prime \prime }+y=15e^{-2t}\), \(y(0)=0\), \(y'(0)=0\), determine \(y(2026)\). Round to the nearest hundredth.
(2)
Determine the “best guess" for a particular solution to the second order linear nonhomogeneous constant-coefficient differential equation
\(y^{\,\prime \prime }+10y^{\,\prime }+25y=te^{5t}\). The phrase “best guess" here means that there is a particular solution of the form given but no particular solution involving fewer terms.
(a)
\(y_p=Ae^{5t}\)
(b)
\(y_p=(At+B)e^{5t}\)
(c)
\(y_p=(At^2+Bt+C)e^{5t}\)
(d)
\(y_p=Ate^{5t}\)
(e)
\(y_p=(At^2+Bt)e^{5t}\)
(f)
\(y_p=(At^3+Bt^2)e^{5t}\)
(3)
The second order linear nonhomogeneous constant-coefficient differential equation \(y^{\,\prime \prime }+10y^{\,\prime }+25y=2e^{-5t}\) does not have a solution of the form \(y=Ae^{-5t}\) where \(A\) is a constant. However, it does have a solution of the form \(ue^{-5t}\) where \(u\) is a non-constant function. What must \(u\) satisfy?
(a)
\(u^{\,\prime }=1\)
(b)
\(u^{\,\prime }=2\)
(c)
\(u^{\,\prime \prime }=1\)
(d)
\(u^{\,\prime \prime }=2\)
(e)
\(u^{\,\prime \prime \prime }=1\)
(f)
\(u^{\,\prime \prime \prime }=2\)
(4)
Consider the 2nd order linear nonhomogeneous equation
\(x^2 y^{\,\prime \prime }+ax y^{\,\prime }+by=cx^d\), where \(a,b,c,d\) are constants and \(c\) is non-zero. Such an equation often has a particular solution of the form \(y_p=Ax^d\) where \(A\) is a non-zero constant. Let \(y(x)\) be such a solution for
\(3x^2 y^{\,\prime \prime }-4xy^{\,\prime }-6y=60x^4\). Determine \(y(1)\). Round to the nearest tenth.
(5)
Let \(a,\omega \) be positive constants. There is a particular solution to
\(y^{\,\prime \prime }+\omega ^2 y=e^{at}\cos {(\omega \, t)}\) of the form \(y_p=e^{at}(A \cos (\omega \, t)+B \sin (\omega \, t))\). Which is \(B\)?
(a)
\(\displaystyle \frac {2\omega }{a^3+4a\omega ^2}\)
(b)
\(\displaystyle \frac {2\omega }{a^2+4\omega ^2}\)
(c)
\(\displaystyle \frac {a}{a^3+4a\omega ^2}\)
(d)
\(\displaystyle \frac {a}{a^2+4\omega ^2}\)
(e)
\(\displaystyle \frac {1}{a^3+4a\omega ^2}\)
(f)
\(\displaystyle \frac {1}{a^2+4\omega ^2}\)