Find y as a function of x if: x^2y"-3xy'-21y=0 y(1)=-8 y'(1)=8 y= find y as a function of x if: x^2y"-5xy'-4y=0 y(1)=2 y'(1)=-1 y=

1)

2)

Step-by-step explanation:

For both of these questions, we will use the Reduction of Orders rule of the Second Order Differential Equations.

1) where and

Let where v is a function of x. Then,

Insert them into the equation,

Choose which eliminates

Hence,

Let , then

Hence,

⇒ ⇒

Now let's take the integral of both sides and then use logarithm rule,

Thus,

Use initial conditions now,

Then and

Finally,

is general solution.

2) where and

Let's use

So roots are

By applying initial conditions, constants are found as follows,

Hence,

is general solution

see attached

step-by-step explanation:

given the function as;

write the equation replacing f(x) with y

form   and fill a table containing values of y with corresponding values of x

you can take a range for   values of x as -3≤x≤3

solving for x=-3

for x=-2

repeat the same procedure to get the values of y for the remaining ,x=-1 to x=3

plot the values of y against values   of x and obtain the following graph