Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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**Converting Non-Linear Equations to Linear Equations** **Objective:** Convert the following non-linear equation into a linear equation using the substitution \( v = y^{1-n} \). Do not solve the equation. \[ y'' - 4y = \frac{8x}{y^2} \] ### Possible Solutions: 1. \( v'' - 12v = 24x \) (Highlighted option) 2. \( v'' - 8v = 16x \) 3. \( v'' - 8v = 24x \) 4. \( v'' + 12v = 12x \) **Explanation:** The substitution method involves transforming a given non-linear differential equation into a linear form through a change of variables. Here, the variable \( v \) is defined as \( v = y^{1-n} \), which simplifies the original equation into a more manageable linear form. The solutions provided offer several transformed variations, with the correct choice marked.
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Transcribed Image Text:**Converting Non-Linear Equations to Linear Equations** **Objective:** Convert the following non-linear equation into a linear equation using the substitution \( v = y^{1-n} \). Do not solve the equation. \[ y'' - 4y = \frac{8x}{y^2} \] ### Possible Solutions: 1. \( v'' - 12v = 24x \) (Highlighted option) 2. \( v'' - 8v = 16x \) 3. \( v'' - 8v = 24x \) 4. \( v'' + 12v = 12x \) **Explanation:** The substitution method involves transforming a given non-linear differential equation into a linear form through a change of variables. Here, the variable \( v \) is defined as \( v = y^{1-n} \), which simplifies the original equation into a more manageable linear form. The solutions provided offer several transformed variations, with the correct choice marked.
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Step 1: Convert the following non linear equation into linear equation
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Step 2: Apply substitution y^3 =v to convert the given equation into linear
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Step 3: Further simplify
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