(D^2+6D+9)y=0,y(0)=2,y(0)=-3

Solution for (D^2+6D+9)y=0,y(0)=2,y(0)=-3 equation:

(^2+6+9)D=0.D(0)=2.D(0)=-3

We move all terms to the left:

(^2+6+9)D-(0.D(0))=0

We add all the numbers together, and all the variables

(^2+6+9)D-0=0

We add all the numbers together, and all the variables

(^2+6+9)D=0

We multiply parentheses

D^2+6D+9D=0

We add all the numbers together, and all the variables

D^2+15D=0

a = 1; b = 15; c = 0;
Δ = b2-4ac
Δ = 152-4·1·0
Δ = 225
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$D_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$D_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{225}=15$
$D_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(15)-15}{2*1}=\frac{-30}{2}
=-15
$
$D_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(15)+15}{2*1}=\frac{0}{2}
=0
$