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Consider the two ODEs dy dx dy dx +y=0 + y² = 0 equil eq.2 (a) How are the two equations different in structure? (b) Show that, for any constant C, the function Ce is always a solution of 1, while Ca-¹ is a solution of 2 only when C = 0 or 1. (c) Show that, for any linear equation of the form dy dx eq:3 if z(x) is a solution, then, for any constant C, the function Cz(x) is also a solution. Hint: Plug Cz(x) into 3. + P(x)y = 0,

Consider the two ODEs dy dx dy dx +y=0 + y² = 0 equil eq.2 (a) How are the two equations different in structure? (b) Show that, for any constant C, the function Ce is always a solution of 1, while Ca-¹ is a solution of 2 only when C = 0 or 1. (c) Show that, for any linear equation of the form dy dx eq:3 if z(x) is a solution, then, for any constant C, the function Cz(x) is also a solution. Hint: Plug Cz(x) into 3. + P(x)y = 0,

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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Consider the two ODES dx dy dx +y=0 + y² = 0 (a) How are the two equations different in structure? (b) Show that, for any constant C, the function Ce is always a solution of 1, while Ca-¹ is a solution of 2 only when C = 0 or 1. (c) Show that, for any linear equation of the form dy dx eqil eq.2 eq.3 if z(x) is a solution, then, for any constant C, the function Cz(x) is also a solution. Hint: Plug Cz(x) into 3. (d) What happens if you try to obtain the same result for an equation of the form dy dx + P(x)y = 0, + P(x)y² = 0? eq.4 Does the same result hold? (e) If you have found one solution of a linear ODE of the form 3, how many solutions do you have? Consult your result from question 2c.
Transcribed Image Text:Consider the two ODES dx dy dx +y=0 + y² = 0 (a) How are the two equations different in structure? (b) Show that, for any constant C, the function Ce is always a solution of 1, while Ca-¹ is a solution of 2 only when C = 0 or 1. (c) Show that, for any linear equation of the form dy dx eqil eq.2 eq.3 if z(x) is a solution, then, for any constant C, the function Cz(x) is also a solution. Hint: Plug Cz(x) into 3. (d) What happens if you try to obtain the same result for an equation of the form dy dx + P(x)y = 0, + P(x)y² = 0? eq.4 Does the same result hold? (e) If you have found one solution of a linear ODE of the form 3, how many solutions do you have? Consult your result from question 2c.
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