A192Appendix AOrdinary Differential Equations: Review of Concepts and Methods5.For the differential equationy-y= 0,p(x) = 0 is its own power series expansionabouta= 0. Soa= 0 is an ordinary point. To solve, lety=∞m=0amxmy=∞m=1mamxm-1y=∞m=2m(m-1)amxm-2.Theny-y=∞m=2m(m-1)amxm-2-∞m=0amxm=∞m=0(m+ 2)(m+ 1)am+2xm-∞m=0amxm=∞m=0[(m+ 2)(m+ 1)am+2-am]xm.Soy-y= 0 implies that(m+ 2)(m+ 1)am+2-am= 0⇒am+2=am(m+ 2)(m+ 1)for allm≥0.Soa0anda1are arbitrary;a2=a02,a4=a24·3=a04!,a6=a46·5=a06!,...a2n=a0(2n)!..Similarly,a2n+1=a1(2n+ 1)!,and soy=a0∞n=01(2n)!x2n+a1∞n=01(2n+ 1)!x2n+1=a0coshx+a1sinhx.9.For the differential equationy+2xy+y= 0,p(x) = 2xis its own power series
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