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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 all the t's XXX XX one side XXX XXX y's on XXXXX.</p><p>XX XXXX XX have,</p><p>dy= (2t +X) dt</p><p>XXX we XXXX XX integrate XXXX XXXXX.</p><p>XXXXX integration we XXX XXX solution XX,</p><p>y=t^2 + 3t + X , XXXXX C is XXX XXXXXXXX of XXXXXXXXXXX.</p><p><br></p><p>XXXXXX one XX <XXXXXX>Integrating XXXXXX Method:<XX></XXXXXX></p><p>Here we XXXX to XXXXX XXXXX the XXXXXXXXXXXX equation in XXX form: dy/dt+p(t)y=X(t)</p><p>Consider an example,</p><p>dy/dt+2y=3</p><p>XXXX p(t)=2 and q(t) =X</p><p>XXX XX XXXX to XXXX an XXXXXXXXXXX XXXXXX i.e. u(t)=e^(XXXXXXXX of p(t) dt)</p><p>i.e. e^{Integral 2dt}= e^(2t)</p><p>XXX we have XX multiply XXX whole XXXXXXXXXXXX XXXXXXXX with the integrating factor. i.e.</p><p>e^(2t) dy/dt+XX^(XX) y=XX^(2t) -----> (1)</p><p>XXX the XXXX XXXX XXXX XXX be written as a XXXXX XXXXXXXXXX i.e.</p><p>e^(2t)dy/dt+XX^(2t)y=(e^(2t)y)'</p><p>Substituting XXXX in (X) we get,</p><p>(e^(XX)y)'=XX^(2t)</p><p>Now integrating both sides we get,</p><p>e^(2t)y=e^(XX)+C</p><p>i.e.</p><p>y=X+XX^(-2t) XX XXX solution with XXXXXXXX X.</p><p><br></p><p><XX></p><p><XX><strong></strong></p><p><br></p><p><br></p><p><br></p><p><br></p><p><XX></p><p><XX></p><p><br></p><p><XX></p><p><XX></p><p><br></p>">
