Given
is a fundamental solution, we posit a second solution of the form
, with derivatives
![{y_2}'=e^{2x}v'+2e^{2x}v=e^{2x}(v'+2v)](https://tex.z-dn.net/?f=%7By_2%7D%27%3De%5E%7B2x%7Dv%27%2B2e%5E%7B2x%7Dv%3De%5E%7B2x%7D%28v%27%2B2v%29)
![{y_2}''=e^{2x}v''+4e^{2x}v'+4e^{2x}v=e^{2x}(v''+4v'+4v)](https://tex.z-dn.net/?f=%7By_2%7D%27%27%3De%5E%7B2x%7Dv%27%27%2B4e%5E%7B2x%7Dv%27%2B4e%5E%7B2x%7Dv%3De%5E%7B2x%7D%28v%27%27%2B4v%27%2B4v%29)
Substitute these into the ODE:
![e^{2x}(v''+4v'+4v)-4e^{2x}(v'+2v)+4e^{2x}v=0\implies v''=0](https://tex.z-dn.net/?f=e%5E%7B2x%7D%28v%27%27%2B4v%27%2B4v%29-4e%5E%7B2x%7D%28v%27%2B2v%29%2B4e%5E%7B2x%7Dv%3D0%5Cimplies%20v%27%27%3D0)
Integrate both sides twice to get
![v''=0\implies v'=C_1\implies v=C_1x+C_2](https://tex.z-dn.net/?f=v%27%27%3D0%5Cimplies%20v%27%3DC_1%5Cimplies%20v%3DC_1x%2BC_2)
Then the second fundamental solution is
![y_2=xe^{2x}+e^{2x}](https://tex.z-dn.net/?f=y_2%3Dxe%5E%7B2x%7D%2Be%5E%7B2x%7D)
but
already cover
, so
.
Taylor has<span> an </span>online bank account<span> that </span>charges<span> a </span>fee<span> of </span>2.3<span>% on </span>each deposit<span>. what is the </span>fee<span> for</span>deposit<span> of </span>62.50<span>? </span>round<span> to the </span>nearest cent<span> - 3585906.</span>
Solution: The commutative property of addition states that when two numbers are being added, their order can be changed without affecting the sum.