d^2y/dx^2−2dy/dx+10y=0 where x=0;y=0 and dy/dx=4?x=0;y=0 and dy/dx=4?

Question

$d^{2} \frac{y}{{d x}^{2}} - 2 \frac{d y}{d x} + 10 y = 0$ where $x = 0; y = 0$ and $\frac{d y}{d x} = 4 ? x = 0; y = 0$ and $\frac{d y}{d x} = 4$ ?

Answer 1

The given differential is $d^{2} \frac{y}{{d x}^{2}} - 2 \frac{d y}{d x} + 10 y = 0$ where $x = 0; y = 0$ and $\frac{d y}{d x} = 4$ .
The auxilary equation is given by $m^{2} - 2 m + 10 = 0$
$\to m = - 2 + (- \sqrt{4 - 40}) \to m = - 2 \pm 2 (\sqrt{1 - 10}) \to m = - 1 \pm \sqrt{1 - 10} \to m = - 1 \pm \sqrt{9} \to m = - 1 \pm 3 i$
Therefore the general olution is given by $y (x) = e^{- x} (c_{1} \cos (3 x) + c_{2} \sin (3 x))$
When $x = 0, y = 0$ therefore $c_{1} = 0$ . Therefore $y = c_{2} e^{- x} (\sin (3 x)) \to d y / x = c_{2} e^{- x} (- \sin (3 x) + 3 \cos (3 x))$
Again when $x = 0, \frac{d y}{d x} = 4$ , this implies that $4 = 3 e 2$
Therefore the solution is $y = (\frac{4 e^{- x}}{3}) \sin (3 x)$

d^2y/dx^2−2dy/dx+10y=0 where x=0;y=0 and dy/dx=4?x=0;y=0 and dy/dx=4?

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