Exercises

(1)
Suppose the following incorrect work is a student’s attempt to use Variation of Parameters to find a solution to \((x-1)y^{\,\prime \prime }-xy^{\,\prime }+y=2(x-1)^2\,e^{x}\) given that \(y_1=x\) and \(y_2=e^x\) form a fundamental set of solutions to the complementary equation. What error is this student making?

The Wronskian of \(y_1=x\) and \(y_2=e^x\) is \(W=y_1y_2^{\,\prime }-y_1^{\prime }y_2=(x-1)e^x\). Then there is a particular solution to \((x-1)y^{\,\prime \prime }-xy^{\,\prime }+y=2(x-1)^2\,e^{x}\) of the form \(y_p=uy_1+vy_2\) where \(\displaystyle u^{\,\prime }=-\frac {2(x-1)^2e^{x}e^x}{(x-1)e^x}=-2(x-1)e^{x}\) and
\(\displaystyle v^{\,\prime }=\frac {2(x-1)^2e^{x}x}{(x-1)e^x}=2x^2-2x\). By integrating, we get \(u=-2(x-2)e^x\) and \(\displaystyle v=\frac {2}{3}x^3-x^2\). So a solution is
\(y_p=ux+ve^x=(-2(x-2)e^x)x+\left (\frac {2}{3}x^3-x^2\right )e^x=\left (\frac {2}{3}x^3-3x^2+4x\right )e^x\).

(a)
Variation of Parameters cannot be used to find a solution to this differential equation.
(b)
The Wronskian of \(y_1=x\) and \(y_2=e^x\) is not \(W=(x-1)e^x\).
(c)
An incorrect “forcing" function \(f(x)\) is being used in the formulas for \(u^{\,\prime }\) and \(v^{\,\prime }\).
(d)
The function \(-2(x-2)e^x\) is not an antiderivative of \(-2(x-1)e^{x}\).
(e)
The function \(\frac {2}{3}x^3-x^2\) is not an antiderivative of \(2x^2-2x\).
(f)
\((-2(x-2)e^x)x+\left (\frac {2}{3}x^3-x^2\right )e^x \neq \left (\frac {2}{3}x^3-3x^2+4x\right )e^x\).
(2)
Correct the error above and determine the correct particular solution \(y_p\) that should have been determined. Determine \(y_p(3)\). Round to the nearest hundredth.
(3)
Suppose \(y(t)\) is the solution to \(t^2 y\,^{\prime \prime }-2y=t^2\) on \((0,\infty )\) which involves the least number of terms. Determine \(y(2)\). Round to the nearest hundredth.
(4)
Using that \(y_1=e^{t}\) and \(\displaystyle y_2=\frac {e^t}{t}\) are a fundamental set of solutions to
\(ty^{\,\prime \prime }-2(t-1)y^{\,\prime }+(t-2)y=0\) on \((0,\infty )\) determine a possible choice for the function \(v\) when using Variation of Parameters to find a particular solution of the form \(y_p=uy_1+vy_2\) for \(ty^{\,\prime \prime }-2(t-1)y^{\,\prime }+(t-2)y=e^{2t}\) on \((0,\infty )\).
(a)
\(v=t\)
(b)
\(v=-0.5e^{2t}\)
(c)
\(v=-e^{-t}\)
(d)
\(v=(t-1)e^t\)
(e)
\(v=-(t-1)e^t\)
(f)
\(v=-e^t\)
(5)
Suppose \(y(x)\) is the particular solution to \(y^{\,\prime \prime }+4y=4\sec ^2(2x)\) which involves the least number of terms. Determine \(\displaystyle y\left (\frac {\pi }{6}\right )\). Round to the nearest hundredth.

Hint: you may find \(\displaystyle \int \sec (\theta )\, \mathrm {d}\theta =\ln \left |\sec (\theta )+\tan (\theta )\right |+C\) helpful.