Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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**Problem Statement:** Find a general solution to the given Cauchy–Euler equation for \( t > 0 \). \[ t^2 \frac{d^2 y}{dt^2} + 2t \frac{dy}{dt} - 6y = 0 \] **Solution Form:** The general solution is \( y(t) = \) [ ] **Explanation:** This problem involves solving a second-order linear homogeneous differential equation with variable coefficients, known as the Cauchy–Euler equation. The structure of this equation is well-suited for solutions involving power functions or exponential functions, typically tackled using substitution methods or by assuming solutions of the form \( y(t) = t^m \), where \( m \) is a constant to be determined.
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Transcribed Image Text:**Problem Statement:** Find a general solution to the given Cauchy–Euler equation for \( t > 0 \). \[ t^2 \frac{d^2 y}{dt^2} + 2t \frac{dy}{dt} - 6y = 0 \] **Solution Form:** The general solution is \( y(t) = \) [ ] **Explanation:** This problem involves solving a second-order linear homogeneous differential equation with variable coefficients, known as the Cauchy–Euler equation. The structure of this equation is well-suited for solutions involving power functions or exponential functions, typically tackled using substitution methods or by assuming solutions of the form \( y(t) = t^m \), where \( m \) is a constant to be determined.
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