Question

Please help me with the below question.

Answer 1

By direct substitution, the polynomic equation y(x) = 1 + a · x + b · x² represents a solution of the differential equation for a = - 2, b = 1/2 and n = 2.

How to analyze a differential equation

In this question we need to analyze a kind of differential equation known of the Laguerre's equation. Differential equations are equations that involves derivatives.

Now we proceed to prove if the expression y(x) = 1 + a · x + b · x² represents a solution of the Laguerre's equation:

$x \cdot \frac{d^{2}y}{dx^{2}}+(1-x) \cdot \frac{dy}{dx} + n\cdot y = 0$     (1)

The proof consists in substituting each term and simplify the resulting expression:

x · (2 · b) + (1 - x) · (2 · b · x + a) + n · (1 + a · x + b · x²) = 0

2 · b · x + 2 · b · x - 2 · b · x² + a - a · x + n + a · n · x + b · n · x² = 0

(- 2 · b + b · n) · x² + (4 · b - a + a · n) · x + (a + n) = 0

The following conditions must be fulfilled:

- 2 + n = 0     (1)

4 · b - a + a · n = 0     (2)

a + n = 0     (3)

By (1) and (3):

n = 2, a = -2

And by (2):

4 · b - (- 2) + (- 2) · (2) = 0

4 · b - 2 = 0

b = 1/2

By direct substitution, the polynomic equation y(x) = 1 + a · x + b · x² represents a solution of the differential equation for a = - 2, b = 1/2 and n = 2.

To learn more on differential equations: brainly.com/question/14620493

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