MATH201 Lecture 26: MATH201(LEC26)
10 Jan 2017
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Example(4:(Find(the(roots(of(the(following(
𝑥!𝑦!! +𝑥𝑥−4𝑦!+6𝑦=0(
Solution:(
1. Compare(the(given(and(the(wanted(equations:(
Want:(𝑥!𝑦!! +𝑥𝑎 𝑥𝑦!+𝑏𝑥𝑦=0(
Given:(𝑥!𝑦!! +𝑥𝑥−4𝑦!+6𝑦=0(
(((((((((((((𝑎𝑥=𝑥−4→𝑎𝑥=𝑎!+𝑎!𝑥+𝑎!𝑥!+𝑎!𝑥!+⋯(
((((((((((((((((((((((((((((((((((((((((→𝑎!=−4, 𝑎!=1, 𝑎!=0 𝑛≥2(
(((((((((((((𝑏𝑥=6→𝑏𝑥=𝑏!+𝑏!𝑥+𝑏!𝑥!+𝑏!𝑥!+⋯ (
(((((((((((((((((((((((((((((((→𝑏0=6, 𝑏!=0 (𝑛≥2)
2. Find(𝑠!(and(𝑠!:(
𝑓𝑠=𝑠!+𝑎!−1𝑠+𝑏!=0(
𝑓𝑠=𝑠!+−4−1𝑠+6=𝑠!−5𝑠+6(
𝑠=
−𝑏±𝑏!−4𝑎𝑐
2𝑎=
5±25 −24
2=
5±1
2=
6
2 𝑜𝑟
4
2(
𝑠!=3 & 𝑠!=2(
3. Find(𝑓(𝑠!+𝑛):(
𝑓𝑠=𝑠!−5𝑠+6(
𝑓𝑠!+𝑛=𝑠!+𝑛!−5𝑠!+𝑛+6(
𝑓3+𝑛=3+𝑛!−53+𝑛+6=9+6𝑛+𝑛!−15 −5𝑛+6=𝑛!+𝑛(
4. Find(𝑐!:(
𝑐!𝑠!=−
1
𝑓𝑠!+𝑛𝑘+𝑠!𝑎!!!+𝑏!!!𝑐!
!!!
!!!
(
= −
1
𝑛𝑛+1𝑘+3𝑎!!!+𝑏!!!𝑐!
!!!
!!!
(
= −
1
𝑛𝑛+1{𝑛−1+3𝑎!!!!!+𝑏!!!!!𝑐!!!
+𝑛−2+3𝑎!!!!!+𝑏!!!!!𝑐!!!+0}(
= −
1
𝑛𝑛+1𝑛+2𝑎!+𝑏!𝑐!!!+𝑛+1𝑎!+𝑏!+0(
=−
1
𝑛𝑛+1{𝑛+2𝑐!!!+0}(
𝑐!𝑠!=−
𝑛+2
𝑛𝑛+1𝑐!!!(
where(
𝑐!=1(
𝑐!=−
1+2
11+1𝑐!=−
3
2(
𝑐!=−
2+2
22+1𝑐!=−
2
3−
3
2=1(
𝑐!=−
3+2
33+1𝑐!=−
5
12 1=−
5
12(
(
(
Document Summary
Example 4: find the roots of the following. Solution: compare the given and the wanted equations: = 0 ( 2: find ! and ! + 4 1 + 6 = ! 15 5 + 6 = : find ! When a differential equation cannot be solved using the power series or the problem specifies to solve using the cauchy-euler method. For of second-order (cauchy) euler equation for y=y(x): + = 0 ( ) where a, b, c are real constants and 0 then ! is a singular point. Compare euler equation to indicial equation and plug-in: 1 + + = 0 ( ) After finding !,! and letting !,! be the root of (*); below are 3 cases: are both real: Both ! and ! are linearly independent solutions of (**) are not real; ie. = !cos () Both ! and ! are linearly independent solutions of (**) Ie. cos ! cos sin ! sin.
