91Ó°ÊÓ

The given parameters are cot $x,1−x\cot x$

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.

$y^{\text{'}\text{'}}−2\left(\csc {\left(x\right)}^2\right)y=\sin {\left(x\right)}^2$

And, given that the homogeneous solution to such differential equation, is given by

$y_1=\cot \left(x\right)\text{ â¶Ä„â¶Ä„}{y}_2=1−x\cot \left(x\right)$

Hence, we can find the particular solution to such differential equation, using the following equation,

$y_p=y_2\left(x\right)∫\frac{y_1\left(x\right)f\left(x\right)}{W\left(x\right)}dx−y_1\left(x\right)∫\frac{y_2\left(x\right)f\left(x\right)}{W\left(x\right)}dx$

the Wronskian is given by

$W\left(x\right)=\left|\begin{array}{ll}{y}_1& y_2\\ y_1^{\text{'}}& y_2^{\text{'}}\end{array}\right|$And, the differentiation of the cosec function is given by,

$\frac{d}{dx}\cot \left(x\right)=−\csc {\left(x\right)}^2$And, hence the Wronskian of the homogeneous equation, is thus

$\begin{array}{l}W\left(x\right)=\left|\begin{array}{cc}\cot \left(x\right)& 1−x\cot \left(x\right)\\ −\csc {\left(x\right)}^2& −\cot \left(x\right)−x\left(−\csc {\left(x\right)}^2\right)\end{array}\right|\\ =\left|\begin{array}{cc}\cot \left(x\right)& 1−x\cot \left(x\right)\\ −\csc {\left(x\right)}^2& −\cot \left(x\right)+x\csc {\left(x\right)}^2\end{array}\right|\\ =−\cot {\left(x\right)}^2+x\csc {\left(x\right)}^2\cot \left(x\right)−\left(−\csc {\left(x\right)}^2+x\cot \left(x\right)\csc {\left(x\right)}^2\right)\\ =−\cot {\left(x\right)}^2+x\csc {\left(x\right)}^2\cot \left(x\right)+\csc {\left(x\right)}^2−x\cot \left(x\right)\csc {\left(x\right)}^2\\ =−\cot {\left(x\right)}^2+\csc {\left(x\right)}^2\\ \begin{array}{l}=−{\left(\frac{\cos \left(x\right)}{\sin \left(x\right)}\right)}^2+\frac{1}{\sin {\left(x\right)}^2}\\ =\frac{−\cos {\left(x\right)}^2}{\sin {\left(x\right)}^2}+\frac{1}{\sin {\left(x\right)}^2}\\ =\frac{−\cos {\left(x\right)}^2+1}{\sin {\left(x\right)}^2}\end{array}\end{array}$

And, knowing the Wronskian of the solutions to the homogeneous equation, and comparing the given differential equation, we find that

$f\left(x\right)=\sin {\left(x\right)}^2$

find the particular solution to the given differential equation,

role="math" localid="1664285934259" $$\begin{gathered} y_p=(1−x\cot (x\left)\right)∫\frac{\cot \left(x\right)\sin {\left(x\right)}^2}{1}dx−\cot \left(x\right)∫\frac{(1−x\cot (x\left)\right)\sin {\left(x\right)}^2}{1}dx \\ \mathrm{Simplifying},\mathrm{the}\mathrm{integration},\mathrm{we}\mathrm{get} \\ \begin{array}{l}{\mathrm{y}}_{\mathrm{p}}=(1−\mathrm{xcot}(\mathrm{x}\left)\right)∫\cot \left(\mathrm{x}\right)\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}−\cot \left(\mathrm{x}\right)∫(1−\mathrm{xcot}(\mathrm{x}\left)\right)\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}\\ {\mathrm{y}}_{\mathrm{p}}=(1−\mathrm{xcot}(\mathrm{x}\left)\right)\underset{{\mathrm{I}}_1}{\underset{ÁåŸ}{∫\cot \left(\mathrm{x}\right)\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}}}−\cot \left(\mathrm{x}\right)∫(1−\mathrm{xcot}(\mathrm{x}\left)\right)\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}\end{array} \\ \mathrm{And},\mathrm{hence}\mathrm{evaluating}\mathrm{the}\mathrm{integral},\mathrm{where} \\ {\mathrm{y}}_{\mathrm{p}}=(1−\mathrm{xcot}(\mathrm{x}\left)\right){\mathrm{I}}_1−\cot \left(\mathrm{x}\right){\mathrm{I}}_2 \\ \mathrm{Thus},\mathrm{evaluating}\mathrm{the}\mathrm{integral}\$\mathrm{I\_}\left\{1\right\}\$,\mathrm{we}\mathrm{get} \\ {\mathrm{I}}_1=∫\cot \left(\mathrm{x}\right)\sin {\left(\mathrm{x}\right)}^2\mathrm{dx} \\ \mathrm{Simplifying}\mathrm{the}\mathrm{integral}, \\ ∫\frac{\cos \left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right)}\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}=∫\cos \left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)\mathrm{dx} \\ \mathrm{hence}\mathrm{the}\mathrm{integration}\mathrm{is}\mathrm{thus} \\ {\mathrm{I}}_1=−\frac{\cos {\left(\mathrm{x}\right)}^2}{2}+\text{const} \\ {\mathrm{I}}_2=∫(1−\mathrm{xcot}(\mathrm{x}\left)\right)\sin {\left(\mathrm{x}\right)}^2\mathrm{dx} \\ \mathrm{Separating}\mathrm{the}\mathrm{integral},\mathrm{we}\mathrm{get} \\ \begin{array}{l}=∫\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}−∫\mathrm{x}\cot \left(\mathrm{x}\right)\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}\\ =∫\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}−∫\mathrm{x}\frac{\cos \left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right)}\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}\\ =∫\sin {\left(\mathrm{x}\right)}^2\mathrm{dx}−∫\mathrm{x}\cos \left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)\mathrm{dx}\end{array} \\ \mathrm{Thus},\mathrm{we}\mathrm{have} \\ \begin{array}{l}=∫\frac{1}{2}(1−\cos (2\mathrm{x}\left)\right)\mathrm{dx}−∫\mathrm{x}\frac{\sin \left(2\mathrm{x}\right)}{2}\mathrm{dx}\\ =\frac{1}{2}∫\mathrm{d}\mathrm{x}−\frac{1}{2}∫\cos \left(2\mathrm{x}\right)\mathrm{dx}−∫\mathrm{x}\frac{\sin \left(2\mathrm{x}\right)}{2}\mathrm{dx}\end{array} \\ \mathrm{Evaluating}\mathrm{the}\mathrm{integral},\mathrm{we}\mathrm{get} \\ \begin{array}{l}{\mathrm{I}}_2=\frac{1}{2}\mathrm{x}−\frac{\sin \left(2\mathrm{x}\right)}{4}−∫\mathrm{x}\frac{\sin \left(2\mathrm{x}\right)}{2}\mathrm{dx}\\ =\frac{\mathrm{x}}{2}−\frac{\sin \left(2\mathrm{x}\right)}{4}−\underset{{\mathrm{I}}_3}{\underset{ÁåŸ}{∫\mathrm{x}\frac{\sin \left(2\mathrm{x}\right)}{2}\mathrm{dx}}}\end{array} \\ \mathrm{Where},\mathrm{the}\mathrm{integral} \\ {\mathrm{I}}_3=∫\mathrm{x}\frac{\sin \left(2\mathrm{x}\right)}{2}\mathrm{dx} \end{gathered}$$

Which can be evaluated by integration by parts, where

Hence, the integration is

$$\begin{gathered} \begin{array}{l}{I}_3=uv−∫vdu\\ =−\frac{x\cos \left(2x\right)}{4}−∫−\frac{\cos \left(2x\right)}{4}dx\\ =−\frac{x\cos \left(2x\right)}{4}+∫\frac{\cos \left(2x\right)}{4}dx\\ =−\frac{x\cos \left(2x\right)}{4}+\frac{\sin \left(2x\right)}{8}\end{array} \\ \begin{array}{l}{I}_2=\frac{1}{2}x−\frac{\sin \left(2x\right)}{4}−\left[−\frac{x\cos \left(2x\right)}{4}+\frac{\sin \left(2x\right)}{8}\right]+\text{const.}\\ =\frac{x}{2}−\frac{\sin \left(2x\right)}{4}+\frac{x\cos \left(2x\right)}{4}−\frac{\sin \left(2x\right)}{8}+\text{const.}\end{array} \\ =\frac{x}{2}−\frac{3\sin \left(2x\right)}{8}+\frac{x\cos \left(2x\right)}{4}+\text{const.} \\ \mathrm{And},\mathrm{from}\left(\mathrm{a}\right)\mathrm{the}\mathrm{particular}\mathrm{solution}\mathrm{is}\mathrm{given}\mathrm{by} \\ {\mathrm{y}}_{\mathrm{p}}=(1−\mathrm{xcot}(\mathrm{x}\left)\right){\mathrm{I}}_1−\cot \left(\mathrm{x}\right){\mathrm{I}}_2 \\ \mathrm{Hence}, \\ \begin{array}{l}={\mathrm{I}}_1−\mathrm{xcot}\left(\mathrm{x}\right){\mathrm{I}}_1−\cot \left(\mathrm{x}\right)\\ {\mathrm{I}}_2=−\frac{\cos {\left(\mathrm{x}\right)}^2}{2}−\mathrm{xcot}\left(\mathrm{x}\right)\left[−\frac{\cos {\left(\mathrm{x}\right)}^2}{2}\right]\end{array} \\ −\cot \left(\mathrm{x}\right)\left[\frac{\mathrm{x}}{2}−\frac{3\sin \left(2\mathrm{x}\right)}{8}+\frac{\mathrm{xcos}\left(2\mathrm{x}\right)}{4}\right] \\ \mathrm{Now},\mathrm{we}\mathrm{try}\mathrm{to}\mathrm{find}\mathrm{a}\mathrm{simpler}\mathrm{form}\mathrm{for}\mathrm{this}\mathrm{particular}\mathrm{solution}, \\ =−\frac{\cos {\left(\mathrm{x}\right)}^2}{2}+\mathrm{xcot}\left(\mathrm{x}\right)\frac{\cos {\left(\mathrm{x}\right)}^2}{2}\mathrm{xcot}\left(\mathrm{x}\right),3\cot \left(\mathrm{x}\right)\sin \left(2\mathrm{x}\right)\text{ â¶Ä„â¶Ä„}\mathrm{xcot}\left(\mathrm{x}\right)\cos \left(2\mathrm{x}\right) \\ \begin{array}{l}−\frac{\mathrm{xcot}\left(\mathrm{x}\right)}{2}+\frac{3\cot \left(\mathrm{x}\right)\sin \left(2\mathrm{x}\right)}{8}−\frac{\mathrm{xcot}\left(\mathrm{x}\right)\cos \left(2\mathrm{x}\right)}{4}\\ =−\frac{\cos {\left(\mathrm{x}\right)}^2}{2}+\mathrm{xcot}\left(\mathrm{x}\right)\left[\frac{\cot {\left(\mathrm{x}\right)}^2}{2}−\frac{\cos \left(2\mathrm{x}\right)}{4}\right]−\frac{\mathrm{xcot}\left(\mathrm{x}\right)}{2}+\frac{3}{8}\cot \left(\mathrm{x}\right)\left(2\sin \right(\mathrm{x}\left)\cos \right(\mathrm{x}\left)\right)\\ =−\frac{\cos {\left(\mathrm{x}\right)}^2}{2}+\mathrm{xcot}\left(\mathrm{x}\right)\left[\frac{\cot {\left(\mathrm{x}\right)}^2}{2}−\frac{\cos \left({\mathrm{x}}^2\right)}{4}+\frac{\sin {\left(\mathrm{x}\right)}^2}{4}\right]−\frac{\mathrm{xcot}\left(\mathrm{x}\right)}{2}+\frac{3}{84}\frac{\cos \left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right)}\left(\cancel{2}\sin \right(\mathrm{x}\left)\cos \right(\mathrm{x}\left)\right)\end{array} \\ \begin{array}{l}=−\frac{\cos {\left(\mathrm{x}\right)}^2}{2}+\mathrm{xcot}\left(\mathrm{x}\right)\left[\frac{\cos \left({\mathrm{x}}^2\right)}{4}+\frac{\sin {\left(\mathrm{x}\right)}^2}{4}\right]−\frac{\mathrm{xcot}\left(\mathrm{x}\right)}{2}+\frac{3}{4}\cos \\ =\cos {\left(\mathrm{x}\right)}^2\left[−\frac{1}{2}+\frac{3}{4}\right]+\frac{\mathrm{xcot}\left(\mathrm{x}\right)}{4}\left[\cos \left({\mathrm{x}}^2\right)+\sin {\left(\mathrm{x}\right)}^2\right]−\frac{\mathrm{xcot}{\left(\mathrm{x}\right)}^2}{2}\\ =\cos {\left(\mathrm{x}\right)}^2\left[\frac{1}{4}\right]+\frac{\mathrm{xcot}\left(\mathrm{x}\right)}{4}\left[1\right]−\frac{\mathrm{xcot}\left(\mathrm{x}\right)}{2}\\ =\frac{\cos {\left(\mathrm{x}\right)}^2}{4}+\mathrm{xcot}\left(\mathrm{x}\right)\left[\frac{1}{4}−\frac{1}{2}\right]\\ =\frac{\cos {\left(\mathrm{x}\right)}^2}{4}−\frac{\mathrm{xcot}\left(\mathrm{x}\right)}{4}\\ =\frac{1}{4}\left(\cos {\left(\mathrm{x}\right)}^2−\mathrm{xcot}\left(\mathrm{x}\right)\right)\\ \mathrm{Thus},\mathrm{the}\mathrm{particular}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{given}\mathrm{differential}\mathrm{equation},\mathrm{is}\mathrm{hence}\\ {\mathrm{y}}_{\mathrm{p}}=\frac{1}{4}\left(\cos {\left(\mathrm{x}\right)}^2−\mathrm{xcot}\left(\mathrm{x}\right)\right)\end{array} \end{gathered}$$uncaught exception: Invalid chunk

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('a2e58aee9f1d6d7...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('