in 
![W(u,v)=\left | \begin{matrix}u&v\\u'&v' \end{matrix} \right |\\=\left | \begin{matrix}e^{\frac{x}{2}}&xe^{\frac{x}{2}}\\\frac{1}{2}e^{\frac{x}{2}}&e^{\frac{x}{2}}\left ( 1+\frac{x}{2} \right ) \end{matrix} \right |\\=e^{\frac{x}{2}}\left [ e^{\frac{x}{2}}\left ( 1+\frac{x}{2} \right ) \right ]-\frac{1}{2}e^{\frac{x}{2}}xe^{\frac{x}{2}}\\=e^x\left ( 1+\frac{x}{2} \right )-\frac{1}{2}xe^x\\=e^x+\frac{1}{2}xe^x-\frac{1}{2}xe^x\\=e^x](/tpl/images/0515/0262/05747.png)
Consider the differential equation
4y'' − 4y' + y = 0; ex/2, xex/2.
Verify t...
Mathematics, 19.02.2020 01:00, georgettemanga2001
Consider the differential equation
4y'' − 4y' + y = 0; ex/2, xex/2.
Verify that the functions
ex/2
and
xex/2
form a fundamental set of solutions of the differential equation on the interval
(−[infinity], [infinity]).
The functions satisfy the differential equation and are linearly independent since W(ex/2, xex/2) =
≠ 0 for
−[infinity] < x < [infinity].
Form the general solution.
y =
Answers: 1
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