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The differential equation y" + p(x)y' + q(x)y = 0 is known as a homogeneous second-order linear equation. It contains y" (second order). The terms y", y' and y all have an exponent of 1 (linear). And the right hand side of the equation is 0 (homogeneous). Prove: if y₁(x) and y₂(x) are two solutions for this equation, then any linear combination of these two functions, Ay₁ (x) + By₂(x), is also a solution.

The differential equation y" + p(x)y' + q(x)y = 0 is known as a homogeneous second-order linear equation. It contains y" (second order). The terms y", y' and y all have an exponent of 1 (linear). And the right hand side of the equation is 0 (homogeneous). Prove: if y₁(x) and y₂(x) are two solutions for this equation, then any linear combination of these two functions, Ay₁ (x) + By₂(x), is also a solution.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The differential equation y" + p(x)y' + q(x)y = 0 is known as a homogeneous second-order linear equation. It contains y" (second order). The terms y", y' and y all have an exponent of 1 (linear). And the right hand side of the equation is 0 (homogeneous). Prove: if y₁(x) and y₂ (x) are two solutions for this equation, then any linear combination of these two functions, Ay₁ (x) + By₂(x), is also a solution.
Transcribed Image Text:The differential equation y" + p(x)y' + q(x)y = 0 is known as a homogeneous second-order linear equation. It contains y" (second order). The terms y", y' and y all have an exponent of 1 (linear). And the right hand side of the equation is 0 (homogeneous). Prove: if y₁(x) and y₂ (x) are two solutions for this equation, then any linear combination of these two functions, Ay₁ (x) + By₂(x), is also a solution.
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