Homework Help is Here – Start Your Trial Now!
Math
Calculus
The General Solution The arbitrary linear combination y = C₁₁ + C₂y₂ of any two solutions with nonzero Wronskian is called the general solution. By the theorem we just proved, we can obtain any particular solution by adjusting C₁ and C₂. We now return to the solutions obtained earlier and prove the final result, for practical purposes the summarizing result. THEOREM 9.3.6 Given the equation y" + ay + by = 0, we form the characteristic equation r²+ ar+b= 0. I. If the characteristic equation has two distinct real roots r₁ and 12, then the general solution takes the form y= C₁e¹¹* + C₂ex II. If the characteristic equation has only one real root r = α, then the general solution takes the form y= C₁ex+C₂xe** = (C₁ + C₂x)e**. III. If the characteristic equation has two complex roots, r₁ = a + iẞ and then the general solution takes the form 12=α- iẞ, y= C₁ex cos ẞx + C₂ex sin ẞx = ex (C₁ cos ẞx+ C₂ sin ẞx). PROOF To prove this theorem, it is enough to show that the three solution pairs ex, ex ραν χρον ex cos ẞx, exsin ẞx all have nonzero Wronskians. The Wronskian of the first pair is the function d d W(x) = e^x(x) - (*) ex dx dx = ex ex-e^x ex = (12-11)(n+1)x W(x) is different from zero since, by assumption, 12 +11. We leave it to you to verify that the other pairs also have nonzero Wronskians. It is time to use what we have learned.

The General Solution The arbitrary linear combination y = C₁₁ + C₂y₂ of any two solutions with nonzero Wronskian is called the general solution. By the theorem we just proved, we can obtain any particular solution by adjusting C₁ and C₂. We now return to the solutions obtained earlier and prove the final result, for practical purposes the summarizing result. THEOREM 9.3.6 Given the equation y" + ay + by = 0, we form the characteristic equation r²+ ar+b= 0. I. If the characteristic equation has two distinct real roots r₁ and 12, then the general solution takes the form y= C₁e¹¹* + C₂ex II. If the characteristic equation has only one real root r = α, then the general solution takes the form y= C₁ex+C₂xe** = (C₁ + C₂x)e**. III. If the characteristic equation has two complex roots, r₁ = a + iẞ and then the general solution takes the form 12=α- iẞ, y= C₁ex cos ẞx + C₂ex sin ẞx = ex (C₁ cos ẞx+ C₂ sin ẞx). PROOF To prove this theorem, it is enough to show that the three solution pairs ex, ex ραν χρον ex cos ẞx, exsin ẞx all have nonzero Wronskians. The Wronskian of the first pair is the function d d W(x) = e^x(x) - (*) ex dx dx = ex ex-e^x ex = (12-11)(n+1)x W(x) is different from zero since, by assumption, 12 +11. We leave it to you to verify that the other pairs also have nonzero Wronskians. It is time to use what we have learned.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 31E
See similar textbooks
icon
Related questions
Question

Explain the key points of Theroem 9.3.6

The General Solution The arbitrary linear combination y = C₁₁ + C₂y₂ of any two solutions with nonzero Wronskian is called the general solution. By the theorem we just proved, we can obtain any particular solution by adjusting C₁ and C₂. We now return to the solutions obtained earlier and prove the final result, for practical purposes the summarizing result. THEOREM 9.3.6 Given the equation y" + ay + by = 0, we form the characteristic equation r²+ ar+b= 0. I. If the characteristic equation has two distinct real roots r₁ and 12, then the general solution takes the form y= C₁e¹¹* + C₂ex II. If the characteristic equation has only one real root r = α, then the general solution takes the form y= C₁ex+C₂xe** = (C₁ + C₂x)e**. III. If the characteristic equation has two complex roots, r₁ = a + iẞ and then the general solution takes the form 12=α- iẞ, y= C₁ex cos ẞx + C₂ex sin ẞx = ex (C₁ cos ẞx+ C₂ sin ẞx).
Transcribed Image Text:The General Solution The arbitrary linear combination y = C₁₁ + C₂y₂ of any two solutions with nonzero Wronskian is called the general solution. By the theorem we just proved, we can obtain any particular solution by adjusting C₁ and C₂. We now return to the solutions obtained earlier and prove the final result, for practical purposes the summarizing result. THEOREM 9.3.6 Given the equation y" + ay + by = 0, we form the characteristic equation r²+ ar+b= 0. I. If the characteristic equation has two distinct real roots r₁ and 12, then the general solution takes the form y= C₁e¹¹* + C₂ex II. If the characteristic equation has only one real root r = α, then the general solution takes the form y= C₁ex+C₂xe** = (C₁ + C₂x)e**. III. If the characteristic equation has two complex roots, r₁ = a + iẞ and then the general solution takes the form 12=α- iẞ, y= C₁ex cos ẞx + C₂ex sin ẞx = ex (C₁ cos ẞx+ C₂ sin ẞx).
PROOF To prove this theorem, it is enough to show that the three solution pairs ex, ex ραν χρον ex cos ẞx, exsin ẞx all have nonzero Wronskians. The Wronskian of the first pair is the function d d W(x) = e^x(x) - (*) ex dx dx = ex ex-e^x ex = (12-11)(n+1)x W(x) is different from zero since, by assumption, 12 +11. We leave it to you to verify that the other pairs also have nonzero Wronskians. It is time to use what we have learned.
Transcribed Image Text:PROOF To prove this theorem, it is enough to show that the three solution pairs ex, ex ραν χρον ex cos ẞx, exsin ẞx all have nonzero Wronskians. The Wronskian of the first pair is the function d d W(x) = e^x(x) - (*) ex dx dx = ex ex-e^x ex = (12-11)(n+1)x W(x) is different from zero since, by assumption, 12 +11. We leave it to you to verify that the other pairs also have nonzero Wronskians. It is time to use what we have learned.
AI-Generated Solution
AI-generated content may present inaccurate or offensive content that does not represent bartleby’s views.
steps

Unlock instant AI solutions

Tap the button
to generate a solution

SEE SOLUTION
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra for College Students
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS