$y''+9y=9sec^{2}3t,0< t< \frac{\pi}{6}\\
solve\,homogeneous\,solution\\
y''+9y=0\\
r^{2}+9=0\\
r=\pm 3i\\
y_{c}(t)=c_{1}cos(3t)+c_{2}sin(3t)\\
solve\,the\,general\,solution\\
replace\,c_{1},c_{2}\,with\,u_{1},u_{2}\\
y(t)=u_{1}(t)cos(3t)+u_{2}(t)sin(3t)\\
u_{1}(t)=-\int \frac{y_{2}(t)g(t)}{w(y_{1},y_{2})(t)}dt+c_{1}\\
u_{2}(t)=-\int \frac{y_{1}(t)g(t)}{w(y_{1},y_{2})(t)}dt+c_{2}\\
y_{1}(t)=cos3t,y_{2}(t)=sin3t,g(t)=9sec^{2}(3t)\\
w(y_{1},y_{2})(t)=3(cos^{2}(3t)+sin^{2}(3t)=3\\
u_{1}(t)=-\int \frac{sin(3t)9sec^{2}(3t)}{3}dt+c_{1}=-sec(3t)+c_{1}\\
u_{2}(t)=-\int \frac{cos(3t)9sec^{2}(3t)}{3}dt+c_{2}=ln(tan(3t)+sec(3t))
+c_{2}\\
y(t)=homogeneous eend ous+particular\\
y(t)=c_{1}cos(3t)+c_{2}sin(3t)+(ln(tan(3t)+sec(3t))+c_{2})sin(3t)-1
w(y_{1},y_{2})(t)=3(cos^{2}(3t)+sin^{2}(3t)=3\\
u_{1}(t)=-\int \frac{sin(3t)9sec^{2}(3t)}{3}dt+c_{1}=-sec(3t)+c_{1}\\
u_{2}(t)=-\int \frac{cos(3t)9sec^{2}(3t)}{3}dt+c_{2}=ln(tan(3t)+sec(3t))
+c_{2}\\
y(t)=homogeneous eend ous+particular\\
y(t)=c_{1}cos(3t)+c_{2}sin(3t)+(ln(tan(3t)+sec(3t))+c_{2})sin(3t)-1
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