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

60: 45 =12: 9 =4: 3 so the answer is 4: 3