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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 XXX on one side XXX all y's on other.</p><p>XX XXXX we XXXX,</p><p>dy= (2t +X) dt</p><p>Now XX XXXX to integrate XXXX sides.</p><p>After XXXXXXXXXXX XX XXX the solution XX,</p><p>y=t^2 + 3t + C , XXXXX C XX XXX XXXXXXXX XX integration.</p><p><br></p><p>Second XXX XX <XXXXXX>XXXXXXXXXXX factor Method:<br></strong></p><p>XXXX we have XX first write the XXXXXXXXXXXX XXXXXXXX in the XXXX: dy/dt+p(t)y=q(t)</p><p>Consider an example,</p><p>dy/dt+XX=3</p><p>XXXX p(t)=X XXX q(t) =X</p><p>Now XX XXXX XX find an XXXXXXXXXXX factor i.e. u(t)=e^(XXXXXXXX of p(t) dt)</p><p>i.e. e^{Integral XXX}= e^(2t)</p><p>XXX we XXXX to multiply XXX XXXXX differential XXXXXXXX XXXX the XXXXXXXXXXX factor. i.e.</p><p>e^(XX) dy/dt+XX^(XX) y=XX^(XX) -----> (1)</p><p>Now XXX XXXX hand side can XX written XX a total XXXXXXXXXX i.e.</p><p>e^(2t)dy/dt+2e^(XX)y=(e^(XX)y)'</p><p>XXXXXXXXXXXX XXXX in (X) we get,</p><p>(e^(XX)y)'=2e^(XX)</p><p>XXX XXXXXXXXXXX both sides XX get,</p><p>e^(XX)y=e^(XX)+X</p><p>i.e.</p><p>y=X+Ce^(-XX) is XXX solution XXXX constant C.</p><p><XX></p><p><XX></p><p><br><strong></XXXXXX></p><p><XX></p><p><XX></p><p><XX></p><p><br></p><p><XX></p><p><br></p><p><XX></p><p><br></p><p><br></p><p><br></p>">
