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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 cross XXXXXXXX so that XXX XXX t's XXX on one XXXX XXX all y's XX other.</p><p>So XXXX XX have,</p><p>dy= (2t +3) dt</p><p>XXX XX XXXX to XXXXXXXXX both sides.</p><p>XXXXX integration we XXX the XXXXXXXX as,</p><p>y=t^2 + 3t + C , where C is the constant XX integration.</p><p><br></p><p>XXXXXX XXX is <XXXXXX>XXXXXXXXXXX factor Method:<br></XXXXXX></p><p>XXXX we XXXX to first write the XXXXXXXXXXXX XXXXXXXX in XXX form: dy/dt+p(t)y=X(t)</p><p>XXXXXXXX an XXXXXXX,</p><p>dy/dt+2y=X</p><p>XXXX p(t)=2 and q(t) =X</p><p>Now XX XXXX to find an integrating XXXXXX i.e. u(t)=e^(Integral XX p(t) dt)</p><p>i.e. e^{Integral 2dt}= e^(2t)</p><p>Now XX XXXX XX XXXXXXXX XXX whole XXXXXXXXXXXX XXXXXXXX XXXX XXX XXXXXXXXXXX XXXXXX. i.e.</p><p>e^(2t) dy/dt+2e^(2t) y=3e^(XX) -----&XX; (1)</p><p>Now XXX left hand side can XX written as a XXXXX XXXXXXXXXX i.e.</p><p>e^(XX)dy/dt+XX^(XX)y=(e^(XX)y)'</p><p>Substituting XXXX in (1) we get,</p><p>(e^(2t)y)'=XX^(2t)</p><p>Now integrating XXXX XXXXX XX XXX,</p><p>e^(2t)y=e^(XX)+C</p><p>i.e.</p><p>y=1+XX^(-2t) XX XXX XXXXXXXX XXXX constant X.</p><p><XX></p><p><XX></p><p><XX><XXXXXX></XXXXXX></p><p><XX></p><p><XX></p><p><XX></p><p><br></p><p><XX></p><p><XX></p><p><XX></p><p><XX></p><p><XX></p><p><XX></p>">
