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6. Consider the equation 3uy + xy = 0. (a) What is its type? (b) Find the general solution. (Hint: Substitute v = Uy.) -3x (c) With the auxiliary conditions u(x, 0) = e−³x and uy(x, 0) = 0, does a solution exist? Is it unique?

6. Consider the equation 3uy + xy = 0. (a) What is its type? (b) Find the general solution. (Hint: Substitute v = Uy.) -3x (c) With the auxiliary conditions u(x, 0) = e−³x and uy(x, 0) = 0, does a solution exist? Is it unique?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 2YT
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[Second Order Equations] How do you solve this?

6. Consider the equation 3uy + xy = 0. (a) What is its type? (b) Find the general solution. (Hint: Substitute v = Uy.) -3x (c) With the auxiliary conditions u(x, 0) = e−³x and uy(x, 0) = 0, does a solution exist? Is it unique?
Transcribed Image Text:6. Consider the equation 3uy + xy = 0. (a) What is its type? (b) Find the general solution. (Hint: Substitute v = Uy.) -3x (c) With the auxiliary conditions u(x, 0) = e−³x and uy(x, 0) = 0, does a solution exist? Is it unique?
Theorem 1. By a linear transformation of the independent variables, the equation can be reduced to one of three forms, as follows. (1) Elliptic case: If a12 < a11922, it is reducible to Uxx + Uyy +... 0 (where... denotes terms of order 1 or 0). (ii) Hyperbolic case: If a²2 > a11922, it is reducible to UxxUyy + = 0. 1.6 TYPES OF SECOND-ORDER EQUATIONS (iii) Parabolic case: If a 12: = a11922, it is reducible to Uxx + = 0 (unless a11 = a12 = a22 = 0). 29
Transcribed Image Text:Theorem 1. By a linear transformation of the independent variables, the equation can be reduced to one of three forms, as follows. (1) Elliptic case: If a12 < a11922, it is reducible to Uxx + Uyy +... 0 (where... denotes terms of order 1 or 0). (ii) Hyperbolic case: If a²2 > a11922, it is reducible to UxxUyy + = 0. 1.6 TYPES OF SECOND-ORDER EQUATIONS (iii) Parabolic case: If a 12: = a11922, it is reducible to Uxx + = 0 (unless a11 = a12 = a22 = 0). 29
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