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Separation of Variables</strong>:</p><p>Consider an ordinary differential equation of the form : dy/dt =2t+3 ,</p><p>Now we take the formula for dy/dt and crXXX XXXXXXXX so that all the t's are on XXX XXXX and all y's XX other.</p><p>So XXXX we have,</p><p>dy= (2t +X) dt</p><p>XXX XX XXXX to XXXXXXXXX both sides.</p><p>After integration XX XXX the XXXXXXXX as,</p><p>y=t^2 + 3t + X , XXXXX X XX XXX constant XX XXXXXXXXXXX.</p><p><br></p><p>Second one is <XXXXXX>XXXXXXXXXXX XXXXXX Method:<XX></strong></p><p>Here we have XX XXXXX XXXXX XXX XXXXXXXXXXXX XXXXXXXX in the XXXX: dy/dt+p(t)y=X(t)</p><p>Consider an example,</p><p>dy/dt+XX=X</p><p>Here p(t)=2 and q(t) =3</p><p>XXX we XXXX XX XXXX an integrating XXXXXX i.e. u(t)=e^(XXXXXXXX of p(t) dt)</p><p>i.e. e^{XXXXXXXX 2dt}= e^(2t)</p><p>XXX XX XXXX XX XXXXXXXX XXX whole XXXXXXXXXXXX XXXXXXXX with the integrating XXXXXX. i.e.</p><p>e^(XX) dy/dt+XX^(XX) y=3e^(XX) -----> (X)</p><p>XXX XXX XXXX hand side XXX be XXXXXXX as a XXXXX derivative i.e.</p><p>e^(2t)dy/dt+XX^(XX)y=(e^(2t)y)'</p><p>XXXXXXXXXXXX this in (1) we get,</p><p>(e^(XX)y)'=2e^(2t)</p><p>Now integrating both XXXXX XX get,</p><p>e^(XX)y=e^(2t)+X</p><p>i.e.</p><p>y=1+XX^(-XX) XX XXX XXXXXXXX XXXX constant C.</p><p><br></p><p><br></p><p><XX><XXXXXX></XXXXXX></p><p><XX></p><p><XX></p><p><br></p><p><XX></p><p><XX></p><p><XX></p><p><XX></p><p><br></p><p><XX></p><p><XX></p>">
