Overview of the Differential Equation
The given differential equation is:
(1+x) * d²y/dx² + x * dy/dx - y = 0
This is a second-order linear ordinary differential equation (ODE) with variable coefficients.
Identifying the Form of the Solution
To solve this type of equation, we typically look for solutions in the form of power series or specific functional forms that satisfy the equation. The presence of the term (1+x) suggests that the equation might involve polynomial solutions or exponential functions.
Potential Solutions
Let's analyze the proposed solutions:
- (a) Ax² + B: This is a polynomial solution of degree 2, which could be a candidate.
- (b) Ax + Be^(-x): This combines a linear term and an exponential decay, which is common for ODEs with variable coefficients.
- (c) Ax + Be^x: This combines a linear term with an exponential growth, which might not satisfy the equation based on the coefficients.
- (d) Ax + Bx²: This is a mix of a linear term and a quadratic term, which might also be a potential solution.
- (e) Ax + Bx²: This is similar to option (d) but lacks the exponential term.
Conclusion
Given the structure of the original differential equation and the form of the coefficients, the most plausible solutions typically involve polynomials or exponentials. Among the options, (b) Ax + Be^(-x) is likely to be the correct form as it incorporates both polynomial and exponential characteristics that can satisfy the behavior dictated by the equation's coefficients.
Thus, the solution is likely to be (b) Ax + Be^(-x).