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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 multiply so XXXX XXX the t's XXX XX XXX XXXX XXX XXX y's on other.</p><p>XX here XX XXXX,</p><p>dy= (XX +X) dt</p><p>XXX XX XXXX to integrate both sides.</p><p>After integration we get the solution as,</p><p>y=t^2 + XX + X , XXXXX X XX XXX constant of integration.</p><p><XX></p><p>XXXXXX XXX is <strong>XXXXXXXXXXX XXXXXX XXXXXX:<br></XXXXXX></p><p>XXXX XX XXXX to XXXXX write XXX differential XXXXXXXX in the form: dy/dt+p(t)y=q(t)</p><p>Consider an XXXXXXX,</p><p>dy/dt+XX=X</p><p>Here p(t)=X XXX X(t) =X</p><p>XXX XX have XX find an integrating XXXXXX i.e. u(t)=e^(Integral of p(t) dt)</p><p>i.e. e^{Integral 2dt}= e^(XX)</p><p>XXX XX have to XXXXXXXX XXX XXXXX XXXXXXXXXXXX XXXXXXXX with XXX XXXXXXXXXXX factor. i.e.</p><p>e^(XX) dy/dt+XX^(XX) y=XX^(XX) -----> (X)</p><p>XXX the left XXXX XXXX XXX XX XXXXXXX XX a total XXXXXXXXXX i.e.</p><p>e^(XX)dy/dt+XX^(2t)y=(e^(XX)y)'</p><p>XXXXXXXXXXXX XXXX in (1) we get,</p><p>(e^(XX)y)'=2e^(XX)</p><p>Now integrating XXXX sides XX XXX,</p><p>e^(2t)y=e^(2t)+C</p><p>i.e.</p><p>y=X+Ce^(-2t) XX the solution with XXXXXXXX X.</p><p><br></p><p><XX></p><p><br><strong></strong></p><p><br></p><p><br></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>">
