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

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

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
7 years ago
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
askIITians Faculty

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