Question

Find a polynomial f(x) satisfying the equation
xf''(x) + 3f(x) = 4x^2.

Answer 1

The question asks for a solution to a differential equation involving finding a polynomial f(x). Solving this typically requires advanced techniques in differential equations. More information or context is needed to provide a unique solution.

Step-by-step explanation:

The student is asking to find a polynomial f(x) that satisfies the differential equation xf''(x) + 3f(x) = 4x². To solve this, one would typically need to apply methods from differential equations to find the function f(x). However, to provide an accurate and complete solution, we need more specific instructions or context since solving such equations often involves integration techniques that are beyond the scope of a general explanation and may require specific boundary conditions or additional information to determine a unique solution.

Answer 2

We're going to assume that the solution is a quadratic polynomial, since we have the sum of the function and its derivative adds up to
`4x^2`. So let
`f(x)=ax^2+bx+c \implies f'(x)=2ax+b \text{ and } f''(x)=2a.`
Plug all these into the given DE to get


`2ax+3ax^2+3bx+3c=4x^2` or

`3ax^2+(2a+3b)x+3c=4x^2`

From the above equation the left hand side should have a coffienet of 4 for
`x^2`, and all other coffienents must be 0. That is

`3a=4,2a+3b=0` and
`3c=0`.

You can easily solve the system above and find
`a=(4)/(3),b=-(8)/(9)` and
`c=0`.

Thus
`f(x)=(4)/(3)x^2-(8)/(9)x`