Show that
and
are linearly independant using the Wronskian and the Gramain (integrate over 1 period)
![{\displaystyle f=cos(7x),g=sin(7x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d7beab3a98af7b5cba6940e421659fc28f8a0e)
One period of ![{\displaystyle 7x=\pi /7}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4278f5c13873c154cd6634f28e43ded580f9e5e1)
Wronskian of f and g
![{\displaystyle W(f,g)=det{\begin{bmatrix}f&g\\f'&g'\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47dfe8cfd362e670f7fcc60915f0ecd5235addbe)
Plugging in values for ![{\displaystyle f,f',g,g';}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46abef6dc84fa8fe963aefdff17a5bf1ba40a1fc)
![{\displaystyle =7cos^{2}(7x)+7sin^{2}(7x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8b0e124ec60f773018903a5ffb0eb1318cb2cf)
![{\displaystyle =7[cos^{2}(7x)+sin^{2}(7x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b301de567d6ca791f8580ee8e9e260a31d103757)
![{\displaystyle =7[1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b92b8a5eea92d540f3a060ab9d154589aa005e48)
They are linearly Independant using the Wronskian.
![{\displaystyle <f,g>=\int _{a}^{b}f(x)g(x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/936d91732c7f2d47bfe4dad4fcca8007e432b661)
![{\displaystyle \Gamma (f,g)=det{\begin{bmatrix}<f,f>&<f,g>\\<g,f>&<g,g>\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91f89b547a18369fe7bf2d7f558a02f5241f26e1)
![{\displaystyle \int _{0}^{\pi /7}cos^{2}(7x)dx=\pi /14}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efba1a93c8bb7b43198d57b0868e58bcf08787f5)
![{\displaystyle \int _{0}^{\pi /7}sin^{2}(7x)dx=\pi /14}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c263bc98e45e25490399794f9aaa94ab3be2956)
![{\displaystyle \int _{0}^{\pi /7}cos(7x)*sin(7x)dx=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44c37749ac7121f5c2e0ae44104e50395189c68e)
![{\displaystyle \Gamma (f,g)=det{\begin{bmatrix}\pi /14&0\\0&\pi /14\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42a7ef16d7aee1475459f1d05dd987c2ef2b73c8)
They are linearly Independent using the Gramain.
Find 2 equations for the 2 unknowns M,N and solve for M,N.
![{\displaystyle y_{p}(x)=Mcos7x+Nsin7x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd1902ad6b6aba1cd6cba7a2ee31e91de27acac)
![{\displaystyle y'_{p}(x)=-M7sin7x+N7cos7x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e042d90da812df623a011eecc813d91e03a29c87)
![{\displaystyle y''_{p}(x)=-M7^{2}cos7x-N7^{2}sin7x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc780d6943b5347afc4a0bdc6710b8b4628969e)
Plugging these values into the equation given (
) yields;
![{\displaystyle -M7^{2}cos7x-N7^{2}sin7x-3(-M7sin7x+N7cos7x)-10(Mcos7x+Nsin7x)=3cos7x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82c56c74161c2d47b0a5b211ec45aa5f9461526d)
Simplifying and the equating the coefficients relating sin and cos results in;
![{\displaystyle -59M-21N=3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a545e0454310c32ee32fe8ebe9ace0f56a8ec25)
![{\displaystyle -59N+21M=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41599723c770b384172beb583ffa875e7933069b)
Solving for M and N results in;
![{\displaystyle M=-177/3922,N=-63/3922}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b165ca4138c70bc7c60c68efce4cc51e06ad49b)
Find the overall solution
that corresponds to the initial conditions
. Plot over three periods.
From before, one period
so therefore, three periods is ![{\displaystyle 3\pi /7.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbee26e8d1cd06196952022decf05b8c3ad165c7)
Using the roots given in the notes
, the homogenous solution becomes;
![{\displaystyle y_{h}(x)=c_{1}e^{-2x}+c_{2}e^{5x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6722fad8e628c9602e377767e8eabae9ab9056c0)
Using initial condtion
;
![{\displaystyle 1=c_{1}+c_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb8e2e309bd352775037bba1b4063e7ce3d5fd3)
with ![{\displaystyle y'(0)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13d7851b0db9f542863ddbbc0f4875874e6bac12)
![{\displaystyle 0=-2c_{1}+5c_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cae60cc88b2e6d1852e7d5b3c46f64b0534f795)
Solving for the constants;
![{\displaystyle c_{1}=5/7,c_{2}=2/7}](https://wikimedia.org/api/rest_v1/media/math/render/svg/231969602e15abd8d16d69ff1b4dcc8228c4ceef)
![{\displaystyle y_{h}(x)=5/7e^{-2x}+2/7e^{5x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05236e08c168c45ea4d34d76d39ccd3abf8476ee)
Using the
found in the last part;
![{\displaystyle y=y_{h}+y_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2756dd440f8cfa429a60bf180c123bdf6bf949c6)
![](//upload.wikimedia.org/wikiversity/en/f/fd/R5_code.jpg)