2Differential equationspower series can someone help me solv

2::::Differential equations:::power series:::: can someone help me solve this please!!

Thanks in advanc

Find the first five nonzero terms in the solution of the given initial value problem. y?? - xy? - y = 0, y(0) = 13, y?(0) = 5 y=

Solution

given differntial equation is

y\'\'-xy\'-y=0 such that y(0)=13,y\'(0)=5

since its a power series, its solution is given by

y(x)== C₀ + C₁(x ? 1) + ((C₀ + C₁) /2) (x ? 1)² + ((C₀ + 3C₁)/ 6) (x ? 1)³ + ((2C₀ + 3C₁) /12) (x ? 1)⁴ + . . .

y(0)=13

y\'(0)=5

y\'\'=xy\'+y

y\'\'\'=xy\'\'+y\'+y\'=xy\'\'+2y\'

y\'\'\'\'=xy\'\'\'+y\'\'+2y\'\'=xy\'\'\'+3y\'\'

so

y\'\'(0)=13

y\'\'\'(0)=10

y\'\'\'\'(0)=39

y(x)== 13 + 5(x ? 1) + 13 (x ? 1)² + 10 (x ? 1)³ + 39 (x ? 1)⁴ + . . .