91Ó°ÊÓ

Variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.

The given equation is:$y^{\text{'}\text{'}\text{'}}+y^{\text{'}}=\sec \mathrm{θ}\tan \mathrm{θ},0<\mathrm{θ}<\mathrm{π}/2$

The auxiliary equation is$m^3+m=0$

$$\begin{gathered} m\left(m^2+1\right)=0 \\ m=0\text{or}{m}^2+1=0 \\ m=0\text{or}m=Â\pm i \end{gathered}$$

The complimentary solution is,$y_c=c_1+c_2\cos \mathrm{θ}+c_3\sin \mathrm{θ}$

The fundamental solution set is$\{1,\cos \mathrm{θ},\sin \mathrm{θ}\}$

$$\begin{gathered} W\left[\begin{array}{ccc}1& \cos \mathrm{θ}& \sin \mathrm{θ}\end{array}\right]=\left|\begin{array}{ccc}1& \cos \mathrm{θ}& \sin \mathrm{θ}\\ 0& -\sin \mathrm{θ}& \cos \mathrm{θ}\\ 0& -\cos \mathrm{θ}& -\sin \mathrm{θ}\end{array}\right| \\ =1 \\ {W}_1\left[\mathrm{θ}\right]=W_1(-1{)}^{3-1}W\left[\begin{array}{cc}\cos \mathrm{θ}& \sin \mathrm{θ}\end{array}\right] \\ =1 \\ {W}_2\left[\mathrm{θ}\right]=(-1{)}^{3-2}W\left[\begin{array}{cc}1& \sin \mathrm{θ}\end{array}\right] \\ =-\left|\begin{array}{cc}1& \sin \mathrm{θ}\\ 0& cos\mathrm{θ}\end{array}\right| \\ {W}_3\left[\mathrm{θ}\right]=(-1{)}^{3-3}W\left[\begin{array}{cc}1& cos\mathrm{θ}\end{array}\right] \\ =-\sin \mathrm{θ} \end{gathered}$$

The particular solution is of the form$y_p\left(x\right)=v_1\left(x\right)\left(1\right)+v_2\left(x\right)\cos \mathrm{θ}+v_3\left(x\right)\left(\sin \mathrm{θ}\right)$

The particular solution is given by:

$$\begin{gathered} y_p=1∫\frac{\sec \mathrm{θ}\tan \mathrm{θ}}{1}d\mathrm{θ}+\cos \mathrm{θ}∫\frac{-\cos \mathrm{θ}}{1}\sec \mathrm{θ}\tan \mathrm{θ}d\mathrm{θ}+\sin \mathrm{θ}∫\frac{-\sin \mathrm{θ}}{1}\left(\sec \mathrm{θ}\tan \mathrm{θ}\right)d\mathrm{θ} \\ =∫\sec \mathrm{θ}\tan \mathrm{θ}d\mathrm{θ}-\cos \mathrm{θ}∫\tan \mathrm{θ}d\mathrm{θ}-\sin \mathrm{θ}∫{\tan }^2\mathrm{θ}d\mathrm{θ} \\ =sec\mathrm{θ}-\cos \mathrm{θ}\ln \left|\sec \mathrm{θ}\right|-\sin \mathrm{θ}(\tan \mathrm{θ}-\mathrm{θ}) \\ =\sec \mathrm{θ}-\cos \mathrm{θ}\ln \left|\sec \mathrm{θ}\right|-\sin \mathrm{θ}\tan \mathrm{θ}+\mathrm{θ}\sin \mathrm{θ} \end{gathered}$$