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Solve d^2y/dt + dy/dt = t^2+2t Given that y(0)=4,y'(0)=2

KHITISH BISWAL , 16 Years ago

Grade 12

1 Answers

Jitender Singh

Ans:

$\frac{d^{2}y}{dt^{2}} + \frac{dy}{dt} = t^{2}+2t$

$\int d(\frac{dy}{dt}) + \int \frac{dy}{dt} = \int (t^{2}+2t)dt$

$(\frac{dy(t)}{dt}) + y(t) = \frac{t^{3}}{3} + t^{2} + c$

$(\frac{dy(0)}{dt}) + y(0) = \frac{(0)^{3}}{3} + (0)^{2} + c$

$4 + 2 = c$

$c = 6$

$(\frac{dy(t)}{dt}) + y(t) = \frac{(t)^{3}}{3} + (t)^{2} + 6$

$e^{t}(y^{'}(t) + y(t)) = e^{t}(\frac{(t)^{3}}{3} + (t)^{2} + 6)$

$\int e^{t}(y^{'}(t) + y(t))dt = \int e^{t}(\frac{(t)^{3}}{3} + (t)^{2} + 6)dt$

$\int (e^{t}.y(t))^{'}dt = \int e^{t}(\frac{(t)^{3}}{3} + (t)^{2} + 6)dt$

$e^{t}.y(t) = \frac{1}{3}.e^{t}(t^{3}+18) + constant$

Thanks & Regards

Jitender Singh

IIT Delhi

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