Question : Find the solution of ode y”-9y’+20y=e^xcos(x) satisfying y(0)=0 y'(0)=0 by variation of parameters method.
Solution :
\[y^{\prime\prime}-9y^\prime+20y=e^x\cos{\left(x\right)} \] (1)
To find complementary solution of ode (1) we solve corresponding homogeneous eqn of (1)
\[y^{\prime\prime}-9y^\prime+20y=0\]
It’s characteristics equation is
\[m^2-9m+20=0\] \[m^2-5m-4m+20=0\]
\[m\left(m-5\right)-4\left(m-5\right)=0\]
\[\left(m-5\right)\left(m-4\right)=0\]
m=4, 5 So
\[y_1\left(x\right)=e^{4x},\ y_{2\left(x\right)}=e^{5x}\]
are fundamental solutions.
Thus complementary solution of ode (1) is
\[y_c\left(x\right)=c_1y_1\left(x\right)+c_2y_2\left(x\right)=e^{4x}c_1+e^{5x}c_2\]
Where \[c_1,\ c_2 \]are arbitrary constants.
Now to find Particular solution by variation of parameters method
Let \[y_p=A\left(x\right)y_1\left(x\right)+B\left(x\right)y_2\left(x\right)\]
\[y_p\left(x\right)=\ e^{4x}A\left(x\right)+e^{5x}B\left(x\right)\]
Where
\[A\left(x\right)=-\int{\frac{y_2\left(x\right)f\left(x\right)}{W(y_1,y_2)}dx}\]
And
\[B\left(x\right)=\int{\frac{y_1\left(x\right)f\left(x\right)}{W(y_1,y_2)}dx}\]
Where Wronksian
\[W\left(y_1.\ y_2\right)=\left|\begin{matrix}y_1&y_2\\y_1\prime&y_2\prime\end{matrix}\right|\]
\[=\left|\begin{matrix}e^{4x}&e^{5x}\\{4e}^{4x}&5e^{5x}\end{matrix}\right|\]
\[=5e^{9x}-4e^{9x}=e^{9x}\]
Hence
\[A\left(x\right)=-\int{\frac{e^{5x}e^x\cos{\left(x\right)}}{e^{9x}}dx}\]
\[=-\int{e^{-3x}\cos{\left(x\right)}dx}\]
\[=-\frac{1}{10}e^{-3x}\left(-3Cos\left(x\right)+Sin\left(x\right)\right)\]
\[B\left(x\right)=\int{\frac{y_1\left(x\right)f\left(x\right)}{W(y_1,y_2)}dx}\]
\[=\int{\frac{e^{4x}e^x\cos{\left(x\right)}}{e^{9x}}dx}\]
\[=\int{e^{-4x}\cos{\left(x\right)}dx}\] \[=\frac{1}{17}e^{-4x}\left(-4Cos\left(x\right)+Sin\left(x\right)\right)\]
Therefore particular solution of ode (1) is
\[y_p\left(x\right)=\ e^{4x}A\left(x\right)+e^{5x}B\left(x\right)\]
\[=-\frac{1}{10}e^x\left(-3Cos\left(x\right)+Sin\left(x\right)\right)+\frac{1}{17}e^x\left(-4Cos\left(x\right)+Sin\left(x\right)\right)\]
\[=\frac{1}{170}e^x\left(11Cos\left(x\right)-7Sin\left(x\right)\right)\]
So general solution of ode (1) is
\[y\left(x\right)=y_c\left(x\right)+y_p\left(x\right)\]
\[y\left(x\right)=e^{4x}c_1+e^{5x}c_2+\frac{1}{170}e^x\left(11Cos\left(x\right)-7Sin\left(x\right)\right)\] (2)
Substituting x = 0 in eqn (2) and using y(0)=0
\[c_1+c_2=-\frac{11}{170}\] (3)
And differentiating eqn (2) with respect to x
\[y^\prime\left(x\right)=4e^{4x}c_1+5e^{5x}c_2+\frac{1}{170}e^x\left(-7Cos\left(x\right)-11Sin\left(x\right)\right)+\frac{1}{170}e^x\left(11Cos\left(x\right)-7Sin\left(x\right)\right)\]
Put x =0 and using y’(0)=0
\[4c_1+5c_2=-\frac{2}{85} \] (4)
Solving eqns (3) and (4)
\[c_1=-\frac{3}{10},\ c_2=\frac{4}{17}\]
Put these values into eqn (2) to get the solution of eqn (1) is
\[y\left(x\right)=-\frac{3}{10}e^{4x}+\frac{4}{17}e^{5x}+\frac{1}{170}e^x\left(11Cos\left(x\right)-7Sin\left(x\right)\right)\]
