The differential equation x22yx2−7xyx+16y=0
has x4 as a solution.
Applying reduction order we set y2=x4.
Then (using the prime notation for the derivatives)
y′2=
y′′2=
So, plugging y2 into the left side of the differential equation, and reducing, we get
x2y′′2−7xy′2+16y2=
The reduced form has a common factor of x5 which we can divide out of the equation so that we have x′′+′=0.
Since this equation does not have any u terms in it we can make the substitution =′ giving us the first order linear equation x′+=0.
This equation has integrating factor
for x > 0.
If we use a as the constant of integration, the solution to this equation is =
Integrating to get u, and using b as our second constant of integration we have =
Finally y2=
and the general solution is

1

step-by-step explanation:

you are throwing one dart

let f (x ) is a function and c be a positive real number,   the graph of the function g (x ) = f (cx ) is represented as follows.

when, c > 1,   the function compresses it in the x - direction.

when, 0 < c < 1, the function stretches it.

the function is g ( x ) = (9x )2.

the heighest power of x in g ( x ) = (9x )2 is 2, so the parent function is square function.

the equation of parent function is f ( x ) = x 2 .

the function is g ( x ) = (9x )2 and the parent function is f ( x ) = x 2 .

so, the functions f and g have the following relationship.

g ( x ) = f ( 9x ).

compare the function g ( x ) = f ( 9x ) with g (x ) = f (cx ).

c = 9 ( > 1 ).

so, the function g (x ) = (9x )2 compresses it in the x - direction.

step-by-step explanation: