# solve the IVP y"+2y'+17y=0.y(0)=1,y'(0)=-1Thanks in advance.Show transcribed image text y'' + 2y' + 17y = 0, y(0) = 1, y'(0) = - 1

solve the IVP $y"+2{y}^{\prime }+17y=0.y\left(0\right)=1,{y}^{\prime }\left(0\right)=-1$

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The characteristic equation is ${r}^{2}+2r+17=0$
The roots of characteristic equation are : $-1±4i$
Then the general solution of homogeneous equation is :.........(i)
Then ${y}^{\prime }\left(0\right)=-{c}_{1}e-t\mathrm{cos}4t-4{c}_{1}e-t\mathrm{sin}4t-{c}_{2}e-t\mathrm{sin}4t-4{c}_{2}e-t\mathrm{cos}4t$ .......(ii)
Using initial values in (i) and (ii), find c1 and c2 and then write the general solution.