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à 1.7è Second Order, Lïear Differential Equations that are
èèèReducible ë First Order
äèèFïd ê general solution
â èèy»» + 2y» = 0èis a lïear second order differential
equation with a missïg y term.èMake ê substitution
p = y» , p» = y»» so ê differential equation becomes
èèp» + 2p = 0.èThis separates ëèdp/p = -2 dx
Integratïg yields ln[p] = -2x - ln[C¬].èRearrangïg gives
èèp = C¬eúì╣.èBut p = y» = C¬eúì╣.èThis ïtegrates ë
ê general solutionèy = -C¬eúì╣ /2 + C½
éS èè A lïear, homogeneous, second order differential
equation is ç ê form
èè a(x)y»» + b(x)y» + c(x)y = 0
If c(x) is ïitially zero, this second order differential
equation can be converted ë two first order differential
equations which can usually be solved by ê techniques ç
this Chapter.
èè For ê differential equation
èè a(x)y»» + b(x)y» = 0
ê substitution
p = y»
åèè p» = y»»
produces ê first order differential equation.
èè a(x)p» + b(x)p = 0
When this is solved for p = G(x), it will have a constant
ç ïtegration.èThis ï turn is substituted back ïë ê
èèèèorigïal variable,
y» = G(x)
å ïtegrated directly ë produce
y = H(x)
This will have two constants ç ïtegration, one from each
ïtegration, which is ê requisite number for a second
order differential equation.
1 y»» + y» = 0
A) C¬x + C½ B) C¬xì + C½
C) C¬e╣ + C½ D) -C¬eú╣ + C½
ü y»» + y» = 0 is missïg its y-term, so make ê
substitution
p = y»
p» = y»»
ë yield
p» + p = 0
This is a separable differential equation which gives
░èdp èè ░è
▒ ────è=è- ▒ dx
▓è p èè ▓è
These ïtegrate ë
ln[p] =è-x + ln[C¬]
Usïg ê properties ç logarithms å rearrangïg yields
p = C¬eú╣
But as p = y», this becomes
y» = C¬eú╣
This ïtegrates directly by substitution
u = -xè du = -dx
ë yield ê general soltuion
y = -C¬eú╣ + C½
ÇèD
2 xìy»» + 2xy» = 0
A) C¬x + C½ B) C¬xì + C½
C) -C¬/x + C½ D) -C¬/xì + C½
ü xìy»» + 2xy» = 0 is missïg its y-term, so make ê
substitution
p = y»
p» = y»»
ë yield
xìp» + 2xp = 0
This is a separable differential equation which gives
░èdp èè ░èdx
▒ ────è= -2 ▒ ────
▓è p èè ▓è x
These ïtegrate ë
ln[p] =è-2ln[x] + ln[C¬]
Usïg ê properties ç logarithms å rearrangïg yields
p = C¬xúì
But as p = y», this becomes
y» = C¬xúì
This ïtegrates directly ë yield ê general solution
y = -C¬xúî + C½
ÇèC
3 4y»» - y» = 0
A) C¬eÅ╣/4 + C½ B) 4C¬e╣»Å + C½
C) -C¬eúÅ╣/4 + C½ D) -4C¬eú╣»Å + C½
ü 4y»» - y» = 0 is missïg its y-term, so make ê
substitution
p = y»
p» = y»»
ë yield
4p» - p = 0
This is a separable differential equation which gives
░èdp è1 ░è
▒ ────è= ─ ▒ dx
▓è p è4 ▓èè
These ïtegrate ë
ln[p] =èx/4 + ln[C¬]
Usïg ê properties ç logarithms å rearrangïg yields
p = C¬e╣»Å
But as p = y», this becomes
y» = C¬e╣»Å
This ïtegrates directly by substitution
u = x/4è du = dx/4è dx = 4du
ë yield ê general solutuion
y = 4C¬e╣»Å + C½
ÇèB
4 xy»» - 3y» = 0
A) C¬xì/2 + C½ B) C¬xÄ/3 + C½
C) C¬xÅ/4 + C½ D) C¬xÉ/5 + C½
ü xy»» - 3y» = 0 is missïg its y-term, so make ê
substitution
p = y»
p» = y»»
ë yield
xp» - 3p = 0
This is a separable differential equation which gives
░èdp èè░èdx
▒ ────è= 3 ▒ ────
▓è p èè▓èx
These ïtegrate ë
ln[p] =è3ln[x] + ln[C¬]
Usïg ê properties ç logarithms å rearrangïg yields
p = C¬xÄ
But as p = y», this becomes
y» = C¬xÄ
This ïtegrates directly ë yield ê general solution
y = C¬xÅ/4 + C½
ÇèC
5 sï[x]y»» - cos[x]y» = 0
A) -C¬cos[x] + C½ B) C¬sï[x] + C½
C) C¬tan[x] + C½ D) -C¬cot[x] + C½
ü sï[x]y»» - cos[x]y» = 0 is missïg its y-term, so
make ê substitution
p = y»
p» = y»»
ë yield
sï[x]p» - cos[x]p = 0
This is a separable differential equation which gives
░èdp è ░ cos[x] dx
▒ ────è=è▒ ─────────
▓è p è ▓è sï[x]
These ïtegrate by substitution on ê right ïtegral
u = sï[x]èèdu = cos[x] dx
ln[p] =èln{sï[x]} + ln[C¬]
Usïg ê properties ç logarithms å rearrangïg yields
p = C¬sï[x]
But as p = y», this becomes
y» = C¬sï[x]
This ïtegrates directly ë yield ê general solution
y = -C¬cos[x] + C½
ÇèA
6 2xy»» - y» = 0
A) 2/3 C¬xÄ»ì + C½ B) 2C¬xî»ì + C½
C) -2C¬xúî»ì + C½ D) -2/3 C¬xúÄ»ì + C½
ü 2xy»» - y» = 0 is missïg its y-term, so make ê
substitution
p = y»
p» = y»»
ë yield
2xp» - p = 0
This is a separable differential equation which gives
░èdp è1 ░èdx
▒ ────è= ─ ▒ ────x
▓è p è2 ▓èx
These ïtegrate ë
ln[p] =è1/2 ln[x] + ln[C¬]
Usïg ê properties ç logarithms å rearrangïg yields
p = C¬xî»ì
But as p = y», this becomes
y» = C¬xî»ì
This ïtegrates directly ë yield ê general solution
y = 2/3 C¬xÄ»ì + C½
ÇèA