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:
[tex]x \cdot \frac{d^{2}y}{dx^{2}}+(1-x) \cdot \frac{dy}{dx} + n\cdot y = 0[/tex] (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: https://brainly.com/question/14620493
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