Mathematics, 24.06.2020 23:01 kaliahgrey

Find the general solution to y′′+6y′+13y=0. Give your answer as y= In your answer, use c1 and c2 to denote arbitrary constants and x the independent variable. Enter c1 as c1 and c2 as c2.

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Answer from: Dashavu4626

$y(x)=c_1e^{-3x} cos(2x)+c_2e^{-3x} sin(2x)$

Step-by-step explanation:

In order to find the general solution of a homogeneous second order differential equation, we need to solve the characteristic equation. This is basically as easy as solving a quadratic.

For a second order differential equation of type:

ay''+by'+cy=0

Has characteristic equation:

$a r^{2} +br+c=0$

Whose solutions r_1 , r_2 ,.., r_n are the roots from which the general solution can be formed. There are three cases:

Real roots:

$y(x)=c_1e^{r_1 x} +c_2e^{r_2 x}$

Repeated roots:

$y(x)=c_1e^{r x} +c_1 xe^{r x}$

Complex roots:

$y(x)=c_1e^{\lambda x}cos(\mu x) +c_2e^{\lambda x}sin(\mu x)\\\\Where:\\\\r_1_,_2=\lambda \pm \mu i$

Therefore:

The characteristic equation for:

y''+6y'+13y=0

Is:

$r^{2} +6r+13=0$

Solving for :

$r_1_,_2= -3 \pm 2i$

So:

$\mu = 2\\\\and\\\\\lambda=-3$

Hence, the general solution of the differential equation will be given by:

$y(x)=c_1e^{-3x} cos(2x)+c_2e^{-3x} sin(2x)$

Answer from: Quest

$y> \frac{1}{3}x$ and y≤ -x +4 .

step-by-step explanation:

we are given a graph for system of inequalities.

in order to write the system of inequalities for the given graph, we need to find the inequality for each of the line separately.

let us find the slope of dotted line first.

taking two coordinates on the graph.

let us say (0,0) and (3,1).

let us find the slope of the line now.

slope of the dotted line would be = $\frac{1-0}{3-0} = \frac{1}{3}$ .

y-intercept of dotted line is y=0 only.

plugging slope m= $\frac{1}{3}$ and b=0 in slope-intercept form y=mx+b, we get

$y=\frac{1}{3}x$ .

the line is shaded up of that dotted line.

therefore, for shading up we would take greater than inequality symbol > .

and because it's a dotted line we would take only > symbol in place of = sign in above equation of line.

therefore, inequality for dotted line would be

$y> \frac{1}{3}x$ .

let us take two points (3,1) and (0,4) on the solid line.

slope of the solid line would be = $\frac{4-1}{0-3} = \frac{3}{-3}=-1$ .

for the solid line we have y-intercept b= 4.

plugging slope m= -1 and b=4 in slope-intercept form y=mx+b, we get

y = -1x + 4.

the line is shaded down of that solid line.

therefore, for shading down we would take less than inequality symbol < .

and because it's a solid line we would take ≤ symbol in place of = sign in above equation of line.

therefore, inequality for solid line would be

y≤ -x +4 .
Write a system of inequalities to represent the shaded portion of the graph.

Answer from: Quest

Im pretty sure it’s is b 1.6 mi

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Find the general solution to y′′+6y′+13y=0. Give your answer as y= In your answer, use c1 and c2 to...