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CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation (4) is said to be linear if F is linear in y, y', ..., y). This means that an nth-order ODE is linear when (4) is an(x)y") + an-1(x)(n-1) + + aj(x)y' + ao(x)y - g(x) = 0 or a₁(x) ● dy Two important special cases of (6) are linear first-order (n = order (n = 2) DES: dx d" y dxn + an-1(x) dn-ly dxn-1 e equations dy + + a₁(x) + ao(x)y= g(x). dx + ao(x)y = g(x) and d'y a₂(x) + a₁(x) dx² dy dx (6) 1) and linear second- In the additive combination on the left-hand side of equation (6) we see that the char- acteristic two properties of a linear ODE are as follows: + ao(x)y = g(x). (7) ?The dependent variable y and all its derivatives y', y", first degree, that is, the power of each term involving y is 1.7 The coefficients ao, a₁, ... , an of y, y', ..., y depend at most on the independent variable x. y(n) are of the

CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation (4) is said to be linear if F is linear in y, y', ..., y). This means that an nth-order ODE is linear when (4) is an(x)y") + an-1(x)(n-1) + + aj(x)y' + ao(x)y - g(x) = 0 or a₁(x) ● dy Two important special cases of (6) are linear first-order (n = order (n = 2) DES: dx d" y dxn + an-1(x) dn-ly dxn-1 e equations dy + + a₁(x) + ao(x)y= g(x). dx + ao(x)y = g(x) and d'y a₂(x) + a₁(x) dx² dy dx (6) 1) and linear second- In the additive combination on the left-hand side of equation (6) we see that the char- acteristic two properties of a linear ODE are as follows: + ao(x)y = g(x). (7) ?The dependent variable y and all its derivatives y', y", first degree, that is, the power of each term involving y is 1.7 The coefficients ao, a₁, ... , an of y, y', ..., y depend at most on the independent variable x. y(n) are of the

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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I am working through differential equations and trying to understand properties of linear ODE as defined in my textbook. From the text in the image is the book trying to say that the coefficients an come from the independent variable x based on the power rule?

CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation (4) is said to be linear if F is linear in y, y', ..., y". This means that an nth-order ODE is linear when (4) is an(x)y") + an-1(x)(n-1)+...+ a₁(x)y' + ao(x)y - g(x) = 0 or a₁(x) dy dx d"y an(x) + an-1(x) dxn Two important special cases of (6) are linear first-order (n = 1) and linear second- order (n 2) DES: = The equations du-ly dxn-1 + ao(x)y = g(x) and dy +・・・ + a₁(x) dx a₂(x) profti. are, in turn lingor f. d²y dx² + ao(x)y = g(x). (y - x) dx + 4x dy = 0, y" - 2y' + y = 0, + a₁(x) visy nob In the additive combination on the left-hand side of equation (6) we see that the char- acteristic two properties of a linear ODE are as follows: dy dx are of the ?The dependent variable y and all its derivatives y', y",. y(n) first degree, that is, the power of each term involving y is 1.? The coefficients ao, a₁, . . . , an of y, y', ..., y depend at most on the independent variable x. and + a₂(x) = g(x). (7) d³y dx³ (6) + x dy dx -- 5y = et
Transcribed Image Text:CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation (4) is said to be linear if F is linear in y, y', ..., y". This means that an nth-order ODE is linear when (4) is an(x)y") + an-1(x)(n-1)+...+ a₁(x)y' + ao(x)y - g(x) = 0 or a₁(x) dy dx d"y an(x) + an-1(x) dxn Two important special cases of (6) are linear first-order (n = 1) and linear second- order (n 2) DES: = The equations du-ly dxn-1 + ao(x)y = g(x) and dy +・・・ + a₁(x) dx a₂(x) profti. are, in turn lingor f. d²y dx² + ao(x)y = g(x). (y - x) dx + 4x dy = 0, y" - 2y' + y = 0, + a₁(x) visy nob In the additive combination on the left-hand side of equation (6) we see that the char- acteristic two properties of a linear ODE are as follows: dy dx are of the ?The dependent variable y and all its derivatives y', y",. y(n) first degree, that is, the power of each term involving y is 1.? The coefficients ao, a₁, . . . , an of y, y', ..., y depend at most on the independent variable x. and + a₂(x) = g(x). (7) d³y dx³ (6) + x dy dx -- 5y = et
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Okay so why is the phrasing "at most" used? It sounds like the coefficents can also be determined by a second variable.

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