Toronto Math Forum

$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
$

${y}''-{2y}'-3y=16cosh(x)
r^2-2r-3=16cosh(x)\\
r^2-2r-3=16(\frac{e^{x}+e^{-x}}{2})=8e^{x}+8e^{-x}\\
(a). Homogeneous part:\\
let\,r^2-2r-3=0\\
=(r+1)(r-3),r_{1}=-1,r_{2}=3\\
y_{c}=C_{1}e^{-x}+C_{2}e^{3x}\\
Next\,we\,solve \, {y}''-{2y}'-3y=8e^{x}\\
let y_{1}=Ae^{x}\\
then\,{y}'=Ae^{x},{y}''=Ae^{x}\\
{y}''-{2y}'-3y=Ae^{x}-2(Ae^{x})-3(Ae^{x})\\
(A-2A-3A)e^{x}=8e^{x}\\
A=-2,y_{1}=-2e^{x}\\
now\,let\,{y}''-{2y}'-3y=8e^{-x}\\
let\, y_{p2}=Bxe^{-x}\,Since Be^{-x}=8e^{-x}\\
then\,{y}'=Be^{-x}-Bxe^{-x},{y}''=-Be^{-x}-Be^{-x}+Bxe^{-x}\\
{y}''-{2y}'-3y=-Be^{-x}-Be^{-x}+Bxe^{-x}-2(Be^{-x}-Bxe^{-x})-3Bxe^{-x}=8e^{-x}\\
B=-2,y_{p2}=-2xe^{-x}\\
y(x)=y_{c}+y_{p1}+y_{p2}=C_{1}e^{-x}+C_{2}e^{3x}-2e^{x}-2xe^{-x}\\
(b)y(x)=C_{1}e^{-x}+C_{2}e^{3x}-2e^{x}-2xe^{-x}\\
y(x)'=-C_{1}e^{-x}+3C_{2}e^{3x}-2e^{x}-2e^{-x}+2xe^{-x}\\
let\,y(0)'=0\\
-C_{1}+3C_{2}-2-2+0=0,-C_{1}+3C_{2}=4\\
let\,y(0)=0\\
C_{1}+C_{2}-2=0,C_{1}+C_{2}=2\\
C_{2}=\frac{2}{3},C_{1}=\frac{1}{2}\\
y=\frac{1}{2}e^{-x}+\frac{2}{3}e^{3x}-2e^{x}-2xe^{-x}$

$The\,Chaacteristic\,eqution\,is:\\
4r^{2}+12r+9=0\\
with\,roots\\
r_{1,2}=\frac{-12\pm \sqrt{144-16*9}}{8}\\
=\frac{-12\pm 0}{8}\\
=-\frac{12}{8}=-\frac{3}{2}\\
Result:y(t)=c_{1}e^{-\frac{3t}{2}}+c_{2}te^{-\frac{3}{2}}$

$Now, \, find \, M_{y}(x,y)\\
=\frac{\delta }{\delta y} (\frac{x}{(x^{2}+y^{2})^{\frac{3}{2}}}\\
=x\frac{\delta }{\delta y}(x^{2}+y^{2})^{\frac{3}{2}}\\
=x(-\frac{3}{2})(x^{2}+y^{2})^{-\frac{5}{2}}(2y)\\
=-(\frac{3yx}{(x^{2}+y^{2})^{\frac{5}{2}}}\\
and \,also, \, N_{x}(x,y)=\frac{\delta }{\delta x}N(x,y)\\

=\frac{\delta }{\delta x} (\frac{y}{(x^{2}+y^{2})^{\frac{3}{2}}})\\
=y\frac{\delta }{\delta x}(x^{2}+y^{2})^{-{\frac{3}{2}}}\\
=y\frac{\delta }{\delta x}{-\frac{3}{2}}((x^{2}+y^{2})^{-{\frac{3}{2}}})\\
=-\frac{3yx}{{x^{2}+y^{2}}^{\frac{2}{5}}}
here, \,observe \,that \,M_{y}(x,y)=N_{x}(x,y)=-\frac{3yx}{({x^{2}+y^{2}})^{\frac{5}{2}}}\\
so \,,the \,given \,differential \,equation \,is \,exact.\\
since \,the \,given \,eqution \,is \,exact \,, then \,there \,exists \,solution,\psi (x,y)such\,that\\
\psi_{x}(x,y)=M(x,y)\\
\psi_{x}(x,y)=\frac{x}{(x^{2}+y^{2})^{\frac{3}{2}}}\\
integrate\,with\,respect\,to\,x, \,then
\int \psi_{x}(x,y)=\int \frac{x}{(x^{2}+y^{2})^{\frac{3}{2}}}dx\\
=\frac{1}{2}\int(x^{2}+y^{2})^{-{\frac{3}{2}}}(2x)dx\\

\frac{1}{2}\frac{({x^{2}+y^{2}})^{-{\frac{1}{2}}}}{-{\frac{1}{2}}}\\
\psi(x,y)=-\frac{1}{{(x^{2}+y^{2})}^{\frac{1}{2}}}+h(y)     \\
(2)\\
next\,find \psi_{y} from (2)\\
differentiate\,it\,with\,respect\,to\,y, \,then \\
\psi_{y}(x,y)=\frac{\delta }{\delta y}(\psi(x,y))\\
=\frac{\delta }{\delta y}(-{\frac{1}{(x^{2}+y^{2})^{\frac{1}{2}}}}+h(y))\\
=-\frac{\delta }{\delta y}{\frac{1}{(x^{2}+y^{2})^{\frac{1}{2}}}}+{\frac{\delta}{\delta y}}h(y)\\
=\frac{1}{{(x^{2}+y^{2})}^{\frac{3}{2}}}+{h}'(y)\\
set\,the\,result\,equal\,to\,N(x,y)\\
\frac{y}{{(x^{2}+y^{2})}^{\frac{3}{2}}}+{h}'(y)=\psi_{x}(x,y)\\
\frac{y}{{(x^{2}+y^{2})}^{\frac{3}{2}}}+{h}'(y)=N(x,y)\\
\frac{y}{{(x^{2}+y^{2})}^{\frac{3}{2}}}+{h}'(y)=\frac{y}{{(x^{2}+y^{2})}^{\frac{3}{2}}}\\
{h}'(y)=0\\
integrate\,with\,respect\,to\,y, then\\
{h}'(y)=0\\
h(y)=0\\
now\,,substitute\,the\,value\,of\,h(y) \,in\,equation(2)\\
\psi(x,y)=-\frac{1}{{(x^{2}+y^{2})}^{\frac{1}{2}}}+h(y)\\
\psi(x,y)=-\frac{1}{{(x^{2}+y^{2})}^{\frac{1}{2}}}+0\\
\psi(x,y)=-\frac{1}{{(x^{2}+y^{2})}^{\frac{1}{2}}}\\
then\,the\,solutions\,of\,equations(1) \,are\,given\,implicitly\,by \\
-\frac{1}{{(x^{2}+y^{2})}^{\frac{1}{2}}}=k\\
{(x^{2}+y^{2})}^{\frac{1}{2}}=\frac{-{1}}{k}\\
x^{2}+y^{2}=\frac{1}{k^{2}}\\
Hence,\,the\,required\,solution\,is\,x^{2}+y^{2}=c,\,here\\
c =\frac{1}{k^{2}}(arbitrary\,constant)$