document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1187735>Question 1187735</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is for differential equations, topic is about separation of variables. <BR>\n" );
document.write( "Find the general and particular solution of the given.\r<BR>\n" );
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document.write( "1.) y' + y = 5sin2x\r<BR>\n" );
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document.write( "2.) y' + 3y = x + e^-2x at x=0 and y=1\r<BR>\n" );
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document.write( "Thank you so much.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #819152 by </B><A HREF=/tutors/aboutme.mpl?userid=Solver92311><B>Solver92311(821)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Solver92311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Solver92311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=819152\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y'\ +\ y\ =\ 5\sin(2x)\">\r<BR>\n" );
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document.write( "First-order linear ODE:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y'(x)\ +\ p(x)y\ =\ q(x)\">\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \mu(x)\ =\ e^{\small{\int\Large \,p(x)\,dx}}\ =\ e^{\small{\int\Large \,1\,dx}}\ =\ e^x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^x\frac{dy}{dx}\ +\ e^xy\ =\ 5e^x\sin(2x)\">\r<BR>\n" );
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document.write( "Note that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large e^x\ =\ \frac{d}{dx}e^x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^x\frac{dy}{dx}\ +\ \frac{d}{dx}e^xy\ =\ 5e^x\sin(2x)\">\r<BR>\n" );
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document.write( "Use the reverse product rule: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f\frac{dg}{dx}\ +\ g\frac{df}{dx}\ =\ \frac{d}{dx}(fg)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}(e^xy)\ =\ 5e^x\sin(2x)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \int\,\frac{d}{dx}(e^xy)\,dx\ =\ \int\,\5e^x\sin(2x)\,dx\">\r<BR>\n" );
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document.write( "Left side:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \int\,\frac{d}{dx}(e^xy)\,dx\ =\ e^xy\">\r<BR>\n" );
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document.write( "Right side:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \int\,\5e^x\sin(2x)\,dx\">\r<BR>\n" );
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document.write( "Integrate by parts twice: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \int\,f\,dg\ =\ fg\ -\ \int\,g\,df\">\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f\ =\ 5\sin(2x)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g\ =\ e^x\">\r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large df\ =\ 10\cos(2x)dx\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large dg\ =\ e^xdx\">, so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \int\,\5e^x\sin(2x)\,dx\ =\ 5e^x\sin(2x)\ -\ \int\,e^x\,10\cos(2x)dx\ +\ C\">\r<BR>\n" );
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document.write( "Integrate by parts a second time to obtain:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \int\,\5e^x\sin(2x)\,dx\ =\ 5e^x\sin(2x)\ -\ 10e^x\cos(2x)\ -\ 4\int\,5e^x\sin(2x)\,dx\ +\ C\">\r<BR>\n" );
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document.write( "Then add <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 4\int\,5e^x\sin(2x)\,dx\"> to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 5\int\,\5e^x\sin(2x)\,dx\ =\ 5e^x\sin(2x)\ -\ 10e^x\cos(2x)\ +\ C\">\r<BR>\n" );
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document.write( "And divide by 5\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \int\,\5e^x\sin(2x)\,dx\ =\ e^x\sin(2x)\ -\ 2e^x\cos(2x)\ +\ C\">\r<BR>\n" );
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document.write( "Therefore:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^xy\ =\ e^x\sin(2x)\ -\ 2e^x\cos(2x)\ +\ C\">\r<BR>\n" );
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document.write( "Divide by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large e^x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \sin(2x)\ -\ 2\cos(2x)\ +\ C\">\r<BR>\n" );
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document.write( "You did not give initial conditions with this one, so there is only a general solution.\r<BR>\n" );
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document.write( "																<BR>\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> <BR>\n" );
document.write( "I > Ø<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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