Thanks in advance for your help

Consider the differential equation x^2 y''-xy'-3y=0. If y1=x3 is one solution use redution of order formula to find a second lin

Anastasy [175]

Suppose $y_2(x)=y_1(x)v(x)$ is another solution. Then

$\begin{cases}y_2=vx^3\\{y_2}'=v'x^3+3vx^2//{y_2}''=v''x^3+6v'x^2+6vx\end{cases}$

Substituting these derivatives into the ODE gives

$x^2(v''x^3+6v'x^2+6vx)-x(v'x^3+3vx^2)-3vx^3=0$

$x^5v''+5x^4v'=0$

Let $u(x)=v'(x)$ , so that

$\begin{cases}u=v'\\u'=v''\end{cases}$

Then the ODE becomes

$x^5u'+5x^4u=0$

and we can condense the left hand side as a derivative of a product,

$\dfrac{\mathrm d}{\mathrm dx}[x^5u]=0$

Integrate both sides with respect to $x$ :

$\displaystyle\int\frac{\mathrm d}{\mathrm dx}[x^5u]\,\mathrm dx=C$

$x^5u=C\implies u=Cx^{-5}$

Solve for $v$ :

$v'=Cx^{-5}\implies v=-\dfrac{C_1}4x^{-4}+C_2$

Solve for $y_2$ :

$\dfrac{y_2}{x^3}=-\dfrac{C_1}4x^{-4}+C_2\implies y_2=C_2x^3-\dfrac{C_1}{4x}$

So another linearly independent solution is $y_2=\dfrac1x$ .

3 0

2 years ago

A circle is the collection of points in a plane that are the same distance from a given point in the plane.

RoseWind [281]

Answer:

True - The above statement is true

A circle is the collection of points in a plane that are the same distance from a given point in the plane.

Step-by-step explanation:

A circle is the set of points in a plane that are a given distance from a given point in the plane. The given point is the center of the circle, and the given distance is the radius.
In other words; A circle is the set of points in a plane that are equidistant from a given center point. Therefore; the locus of points at a fixed distance, d, from the point, P is a circle with the given point P as its center and d as its radius.

6 0

3 years ago

Which phrases can be used to represent the inequality 6.5x+1.5<21? Select two options.

guapka [62]

Answer:

Phrases 2 and 4.

Step-by-step explanation:

6.5x+1.5<21

The sum of 1.5 and the product of 6.5 and a number is no greater than 21.

The product of 6.5 and a number, when increased by 1.5 is at most 21.

HOPE IT HELPS

(FROM CROSS)

6 0

2 years ago

Need help on these problems

kupik [55]

Simple Interest = Principal * Interest Rate * Term
Total Amount = Principal + Simple Interest

 # Principal Interest rate term Interest Total balance
 1 100.00 6% 1 6.00 106.00
 3 100.00 6% 2 12.00 112.00
 5 100.00 6% 3 18.00 118.00
 7 100.00 6% 4 24.00 124.00
 9 3,000.00 5% 3 450.00 3,450.00
10 4,000.00 6% 3 720.00 4,720.00
11 10,000.00 4% 3 1,200.00 11,200.00
13 10,000.00 6% 4 2,400.00 12,400.00
14 10,000.00 6% 5 3,000.00 13,000.00

5 0

2 years ago

Identifying equivalent statements and negations of a conditional statement: help

Vitek1552 [10]

Answer:

1: Equal

2: Negation

3: Negation

4: Neither

Step-by-step explanation:

Sorry if I got any of them wrong

6 0

2 years ago

Thanks in advance for your help ​

Thanks in advance for your help