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use-power-series-solution-method-to-solve-the-ODE-y-xy-0-




Question Number 99646 by 24224 Opiyo Kamuki last updated on 22/Jun/20
use power series solution method to solve the ODE  y′′−xy=0
usepowerseriessolutionmethodtosolvetheODEyxy=0
Answered by MWSuSon last updated on 22/Jun/20
Σ_(k=2) ^∞ k(k−1)a_k x^(k−2) −xΣ_(k=0) ^∞ a_k x^k =0  Σ_(k=0) ^∞ (k+1)(k+2)a_(k+2) x^k −Σ_(k=0) ^∞ a_k x^(k+1) =0  Σ_(k=0) ^∞ (k+1)(k+2)a_(k+2) x^k −Σ_(k=1) ^∞ a_(k−1) x^k =0  2a_2 +Σ_(k=1) ^∞ [(k+2)(k+1)a_(k+2) −a_(k−1) ]x^k =0  a_2 =0  recurrence relation for k≥1  a_(k+2) =(a_(k−1) /((k+2)(k+1)))  input values for k=1,2,3,4,5,...  and find a_k  interms of a_(0 ) and a_1
k=2k(k1)akxk2xk=0akxk=0k=0(k+1)(k+2)ak+2xkk=0akxk+1=0k=0(k+1)(k+2)ak+2xkk=1ak1xk=02a2+k=1[(k+2)(k+1)ak+2ak1]xk=0a2=0recurrencerelationfork1ak+2=ak1(k+2)(k+1)inputvaluesfork=1,2,3,4,5,andfindakintermsofa0anda1
Answered by mathmax by abdo last updated on 22/Jun/20
y =Σ_(n=0) ^∞  a_n x^n  ⇒y^′  =Σ_(n=1) ^∞  na_n x^(n−1)  ⇒y^(′′)  =Σ_(n=2) ^∞  n(n−1)a_n x^(n−2)   =Σ_(n=0) ^∞  (n+2)(n+1)a_(n+2)  x^n   e⇒Σ_(n=0) ^∞  (n+1)(n+2)a_(n+2) x^n  −Σ_(n=0) ^∞  a_n x^(n+1)  =0 ⇒  Σ_(n=0) ^∞ (n+1)(n+2)a_(n+2) x^n  −Σ_(n=1) ^∞  a_(n−1) x^(n  )  =0 ⇒  2a_(2 )  +Σ_(n=1) ^∞ {(n+1)(n+2)a_(n+2) −a_(n−1) }x^n  =0 ⇒  a_2 =0 and  a_(n+2) =(a_(n−1) /((n+1)(n+2)))  (n≥1) ⇒a_3 =(a_0 /(2.3))  a_4 =(a_1 /(3.4))  ,    a_5 =(a_2 /(4.5)) ,.... ⇒y(x) =a_0  +a_1 x +a_2 x^2  +a_3 x^3  +a_4 x^4  +...  =a_0  +a_1 x +(a_0 /6) x^3  +(a_1 /(12))x^4  +(a_2 /(20)) x^5  +...
y=n=0anxny=n=1nanxn1y=n=2n(n1)anxn2=n=0(n+2)(n+1)an+2xnen=0(n+1)(n+2)an+2xnn=0anxn+1=0n=0(n+1)(n+2)an+2xnn=1an1xn=02a2+n=1{(n+1)(n+2)an+2an1}xn=0a2=0andan+2=an1(n+1)(n+2)(n1)a3=a02.3a4=a13.4,a5=a24.5,.y(x)=a0+a1x+a2x2+a3x3+a4x4+=a0+a1x+a06x3+a112x4+a220x5+

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