Solve the differential equation: d2ydx2−8dydx=−16y? Discuss what kind of differential equation is this, and when it may arise?

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Pubephenedsjq · Accepted Answer

d2ydx2−8dydx=−16ybest written asd2ydx2−8dydx+16y=0which shows that this is linear second order homogeneous differential equationit has characteristic equationr2−8r+16=0which can be solve as follows(r−4)2=0,r=4this is a repeated root so the general solution is in formy=(Ax+B)e4xthis is non-oscillating and models some kind of exponential behaviour that really depends on the value of A and B. One might guess it could be an attempt to model population or predator/prey interaction but i can&#039;t really say anything very specific.

diocedss33 · Accepted Answer

The differential equationd2ydx2−8dydx+16y=0is a linear homogeneous constant coefficient equation.For those equations the general solution has the structurey=eλxSubstituting we haveeλx(λ2−8λ+16)=0Here eλx≠0 so the solutions must obeyλ2−8λ+16=(λ−4)2=0Solving we obtainλ1=λ2=4When the roots repeat, ddλeλx is also solution. In case of n roots repeated, we will have as solutions:Cididλieλx for i=1,2,.., nSo, to maintain the number of initial conditions, we include them as independent solutions.In this case we havey=C1eλx+C2ddλeλxwhich results iny=(C1+C2x)eλx

Solve the differential equation: \(\displaystyle{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}-{8}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}=-{16}{y}\)? Discuss what

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