Pro One: Differential Equations

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à 4.4 NON-HOMOGENEOUS EQUATIONS; VARIATION OF PARAMETERS äèFïd ê particular solution, given ê fundmental èèèèèèèset ç solutions, ë ê non-homogeneous equation. â è Forèy»» - 5y» + 6y = 2x + 3 The fundamental solutions set is eì╣ å eÄ╣. Thus assume a particular solution ç ê formè u¬(x)eì╣ + u½(x)eÄ╣ Solvïg yieldsèu¬ = eúì╣(x + 2) u½ = eúÄ╣(-2x/3 + 11/9) The particular solution isèx/3 + 7/9 éS è The problem ç solvïg a lïear, NON-HOMOGENEOUS, second order differential equation can be split ïë two parts 1) Solve ê homogeneous differential equation which will produce two ïdependent solutions (ê fundamental set ç solutions) say y¬ å y½.èThe general solution ç ê homogeneous differential equation is C¬y¬ + C½y½ 2) Fïd any PARTICULAR SOLUTION ë ê NON-HOMGENEOUS differential equation, say y╞. èèThe general solution ë ê NON-HOMOGENEOUS differential equation is ê sum ç êse two solutions C¬y¬ + C½y½ + y╞ èè In this section, ê METHOD OF VARIATION OF PARAMETERS will be used ë fïd PARTICULAR SOLUTIONS ç ê NON-HOMOGEN- EOUS differential equation.èThis method is one that will work ï general.èIts only restriction is ê ability ë evaluate two ïtegrals.èAssumïg ability ë ïtegrate êse ïtegrals a particular solution can always be found. èè If ê non-homogeneous term is a) a polynomial b) a real exponential c) a real exponential times sï[x] or cos[x] Or a lïear combïation ç êse, ê METHOD OF UNDETERMINED COEFFICIENTS Section 5.3) can be used MORE EASILY ë fïd a particular solution. èè The method ç VARIATION OF PARAMETERS assumes that ê form ç ê particular solution is as follows, y = u¬(x)y¬ + u½(x)y½, where y¬ å y½ form ê fundamental solution set ç ê homogeneous differential equation å u¬ å u½ are functions ç x that will be determïed. As êre are TWO unknown functions u¬ å u½, TWO EQUATIONS are needed ë solve for êm.èOne will be obtaïed by direct substituion ç ê assumed solution ïë ê NON-HOMO- GENEOUS differential equation.èThe second condition was developed by LAGRANGE who developed this method.è èèèDifferentiatïg ê assumed solution yields y» = u¬»y¬ + u¬y¬» + u½»y½ +u½y½» LaGrange's condition was that ê first å third terms add ë zero u¬»y¬ + u½»y½ = 0 èè This leaves ê first derivative as y» = u¬y¬» + u½y½» èè Differentiatïg agaï å substitutïg ïë ê non- homogeneous diferential equation å recallïg that y¬ å y½ are solutions ç ê homogeneous differential equation yield ê second condition that u¬»y¬» + u½»y½» = g(x) Thus we have ê system ç equations u¬»y¬è+ u½»y½è= 0 u¬»y¬» + u½»y½» = g(x) ï ê two unknowns u¬» å u½». èè Solvïg this system by use ç Cramer's Rule yields y½(x)g(x) u¬» = - ─────────── èW(y¬;y½) y¬(x)g(x) u½» =è ─────────── èW(y¬;y½) where W(y¬;y½) is ê WRONKSIAN ç y¬ å y½ explaïed ï Section 5.1 å is given by è ▒èy¬è y½è│ W(y¬;y½) = ▒èèèèè │è=è y¬y½» - y¬»y½ è ▒èy¬»èy½» │ èè The two functions u¬ å u½ can now be found by ïtegration ░èy½(x)g(x) u¬ = -è▒ ─────────── dx ▓èW(y¬;y½) ░èy¬(x)g(x) u½ =èè▒ ─────────── dx ▓è W(y¬;y½) The only obstacle ï this method lies ï ê ability ë evaluate êse ïtegrals.èWhen this is done, ê particular solution is y = u¬(x)y¬ + u½(x)y½ å ê general solution is y = C¬y¬ + C½y½ + u¬(x)y¬ + u½(x)y½ 1èèy»» + 4y» + 3y = 2xèFundamental set are eú╣ å eúÄ╣ A) 2/3 x + 8/9 B) 2/3 x - 8/9 C) -2/3 x + 8/9 D) -2/3 x - 8/9 ü The Wronskian ç ê fundamental solution set is èè │èeú╣èeúÄ╣è│ W =è│èèèèèè │ èè │ -eú╣ -3eúÄ╣ │ è=è-3eú╣eúÄ╣ + eú╣eúÄ╣ è=è-2eúÅ╣ èèè ░èeúÄ╣ (2x) u¬ = - ▒ ─────────── dx èèè ▓è -2eúÅ╣ èèè ░ è =è ▒ x e╣ dx èèè ▓ This is ïtegrated by parts usïg u = xèsoèdu = dx dv = e╣dxèsoèv = e╣ è ░ u¬ = xe╣ - ▒ e╣ dx è ▓ which ïtegrates ë u¬ = xe╣ - e╣ Similarly èèè░èeú╣ (2x) u½ =è▒ ────────── dx èèè▓è -2eúÅ╣ èèèè░ è =è- ▒ x eÄ╣ dx èèèè▓ This is ïtegrated by parts usïg u = xèsoèdu = dx dv = eÄ╣dxèsoèv = eÄ╣/3 èèè ░ u½ = -xeÄ╣/3 + ▒ eÄ╣/3 dx èèè ▓ which ïtegrates ë u½ = - xeÄ╣/3 + eÄ╣/9 The particular solution becomes y╞ = (xe╣ - e╣)eú╣ + (- xeÄ╣/3 + eÄ╣/9)eúÄ╣ è =èx - 1 - x/3 + 1/9 è =è2/3 xè-è8/9 The general solution is C¬eú╣ + C½eúÄ╣ + 2/3 xè- 8/9 ÇèB 2èy»» - 4y» + 3y = 5eÄ╣ ; Fundamental set are e╣ å eÄ╣ A) 5/2 xeÄ╣ B) 5/2 eÄ╣ C) -5/2 xeÄ╣ D) -5/2 eÄ╣ ü The Wronskian ç ê fundamental solution set is èè │èe╣èèeÄ╣è│ W =è│èèèèèè │ èè │èe╣è 3eÄ╣è│ è=è3e╣eÄ╣ - e╣eÄ╣ è=è2eÅ╣ èèè ░èeÄ╣ (5eÄ╣) u¬ = - ▒ ─────────── dx èèè ▓èè 2eÅ╣ èèè 5 ░ è = - ─ ▒èe║╣ dx èèè 2 ▓ This is ïtegrated directly ë u¬ =è-5/4 eì╣ Similarly èèè░èe╣ (5eÄ╣) u½ =è▒ ────────── dx èèè▓èè2eÅ╣ èèè5 ░ è =è─ ▒ dx èèè2 ▓ This is ïtegrated directly ë u½ = 5/2 x The particular solution becomes y╞ = (-5/4 eì╣)e╣ + (5/2 x)eÄ╣ è =è5/2 x eÄ╣ - 5/4 eÄ╣ As eÄ╣ is part ç ê fundamental set, ê second term is redundant ï ê general solution so ê particular solution is, most simply è =è5/2 x eÄ╣ The general solution is C¬eú╣ + C½eúÄ╣ + 5/2 x eÄ╣ ÇèA 3èy»» + y = sï[x]èFundamental set are cos[x] å sï[x] A) 1/2 x sï[x] B) 1/2 x cos[x] C) -1/2 x sï[x] D) -1/2 x cos[x] ü The Wronskian ç ê fundamental solution set is èè │ècos[x]è sï[x]è│ W =è│èèèèèèèèè │ èè │ -sï[x]è cos[x]è│ è=ècosì[x] + sïì[x] è=è1 èèè ░èsï[x] (sï[x]) u¬ = - ▒ ───────────────── dx èèè ▓èèèè 1 èèè ░ è = - ▒ sïì[x] dx èèè ▓ This is ïtegrated by double angle substituion ░è1 - cos[2x] u¬ =è- ▒ ───────────── dx ▓èèè 2 which ïtegrates u¬ = -x/2 + sï[2x]/4 Usïg ê double angle identity u¬ = -x/2 + sï[x]cos[x]/2 Similarly èèè░ècos[x] (sï[x]) u½ =è▒ ───────────────── dx èèè▓èèèè 1 èèèè░ è =èè▒ sï[x] cos[x] dx èèèè▓ This is ïtegrated by substituion usïg u = sï[x]èsoèdu = cos[x] dx èèè░ u½ =è▒ u du èèè▓ which ïtegrates ë u½ = uì/2 Changïg back ë ê origïal ïtegration variable u½ = sïì[x]/2 The particular solution becomes èè y╞ = (-x/2 + sï[x]cos[x]/2)cos[x] + ( sïì[x]/2)sï[x] è =è-x cos[x]/2 + sï[x]/2 {così[x] + sïì[x]} è =è-x cos[x]/2 + sï[x]/2è As sï[x] is part ç ê fundamental set, ê second term is redundant ï ê general solution so ê particular solution is, most simply y╞ = - 1/2 x cos[x] The general solution is C¬cos[x] + C½sï[x] - 1/2 x cos[x] ÇèD 4èèx║y»» - 6y = 2x + 4èFundamental set are xÄ å xú║ A) 2/5 xÄln[x] + xì B) 2/5 xÄln[x] - xì C) - 2/5 xÄln[x] + xì D) - 2/5 xÄ ln[x] - xì ü The Wronskian ç ê fundamental solution set is èè │è xÄèèxúìè │ W =è│èèèèèèè │ èè │è3xìè -2xúÄè│ è=è-2xÄxúÄ - 3xìxúì è=è-5 èèè ░èxúì (2x+4) u¬ = - ▒ ──────────── dx èèè ▓èèè-5 èè 2è░èdxèè 4è░èdx è = ─è▒ ────è+ ─è▒ ──── èè 5è▓è xèè 5è▓èx║ This is ïtegrated by ê power rule u¬ = 2/5 ln[x] - 4/5 xúî Similarly èèè░èxÄ (2x+4) u½ =è▒ ─────────── dx èèè▓èèè5 èèè 2è░èèèèè4è░ è = - ─è▒ xÅ dxè- ─è▒ xÄ dx èèè 5è▓èèèèè5è▓ This is ïtegrated by ê power rule u½ =è- 2/25 xÉ - 1/5 xÅ The particular solution becomes èè y╞ = (2/5 ln[x] - 4/5 xúî)xÄ è+ (- 2/25 xÉ - 1/5 xÅ)xúì è =è2/5 xÄ ln[x] - 4/5 xì - 2/25 xÄ - 1/5 xì è =è2/5 xÄ ln[x] - 2/25 xÄ -èxì As xÄ is part ç ê fundamental set, ê second term is redundant ï ê general solution so ê particular solution is, most simply y╞ = 2/5 xÄ ln[x] - xì The general solution is C¬xÄ + C½xúìè+ 2/5 xÄ ln[x] - xì ÇèB 5èy»» + y = cot[x]èFundamental set are cos[x] å sï[x] A) sï[x] ln│csc[x] + cot[x]│ B) -sï[x] ln│csc[x] + cot[x]│ C) cos[x] ln│csc[x] + cot[x]│ D) -cos[x] ln│csc[x] + cot[x]│ ü The Wronskian ç ê fundamental solution set is èè │ècos[x]è sï[x]è│ W =è│èèèèèèèèè │ èè │ -sï[x]è cos[x]è│ è=ècosì[x] + sïì[x] è=è1 èèè ░èsï[x] (cot[x]) u¬ = - ▒ ───────────────── dx èèè ▓èèèè 1 èèè ░èèèè cos[x] è = - ▒ sï[x] ──────── dx èèè ▓èèèè sï[x] èèè ░èèèè è = - ▒ cos[x] dx èèè ▓èèèè This is ïtegrated directly ë u¬ = -sï[x] Similarly èèè░ècos[x] (cot[x]) u½ =è▒ ───────────────── dx èèè▓èèèè 1 èèè░ èèèècos[x] è =è▒ cos[x] ──────── dx èèè▓ èèèèsï[x] èèè░ècosì[x] è =è▒ ───────── dx èèè▓èsï[x] èèè░è1 -sïì[x] è =è▒ ──────────── dx èèè▓èèsï[x] èèè░è è =è▒ csc[x] - sï[x] dx èèè▓è which ïtegrates ë u½ =è- ln │ csc[x] + cot[x] │ + cos[x] The particular solution becomes èè y╞ = (- sï[x])cos[x] + (-ln│ csc[x] + cot[x] │ + cos[x])sï[x] è =è-sï[x] cos[x] + sï[x] ln│ csc[x] + cot[x] │ + sï[x] cos[x] è =è sï[x] ln│ csc[x] + cot[x] │ The general solution is C¬cos[x] + C½sï[x] + sï[x]ln│ csc[x] + cot[x] │ ÇèA