- #1
ChrisVer
Gold Member
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I have to solve the differential equation
[itex] y''+(1-t) y' + y= sin(2t)[/itex]
can someone judge this?
How could I continue it?
[itex] y=\sum_{n=0}^{∞}{a_{n} t^{n}}[/itex]
[itex] y'=\sum_{n=1}^{∞}{a_{n} n t^{n-1}}[/itex]
[itex] y''=\sum_{n=2}^{∞}{a_{n} n(n-1) t^{n-2}}[/itex]
[itex] sin(2t)=\sum_{n=0}^{∞}{\frac{2^{2n}}{2n!} t^{2n}}[/itex][itex] y''+(1-t) y' + y= sin(2t)[/itex]
[itex] \sum_{n=2}^{∞}{a_{n} n(n-1) t^{n-2}}+\sum_{n=1}^{∞}{a_{n} n (1-t)t^{n-1}}+\sum_{n=0}^{∞}{a_{n} t^{n}}=\sum_{n=0}^{∞}{\frac{2^{2n}}{2n!} t^{2n}}[/itex]
[itex] \sum_{n=2}^{∞}{a_{n} n(n-1) t^{n-2}}+\sum_{n=1}^{∞}{a_{n} n t^{n-1}}-\sum_{n=1}^{∞}{a_{n} n t^{n}}+\sum_{n=0}^{∞}{a_{n} t^{n}}=\sum_{n=0}^{∞}{\frac{2^{2n}}{2n!} t^{2n}}[/itex]
[itex] \sum_{k=0}^{∞}{a_{k+2} (k+2)(k+1) t^{k}}+\sum_{k=0}^{∞}{a_{k+1} (k+1) t^{k}}-\sum_{k=0}^{∞}{a_{k} k t^{k}}+\sum_{k=0}^{∞}{a_{k} t^{k}}=\sum_{k=0}^{∞}{\frac{2^{2k}}{2k!} t^{2k}}[/itex]
[itex] \sum_{k=0}^{∞}{(a_{k+2} (k+2)(k+1)+a_{k+1} (k+1) -a_{k} k + a_{k}) t^{k}}=\sum_{k=0}^{∞}{\frac{2^{2k}}{2k!} t^{2k}}[/itex]
[itex] \sum_{k=0}^{∞}{(a_{k+2} (k+2)(k+1)+a_{k+1} (k+1)+ (1-k) a_{k}) t^{k}}=\sum_{k=0}^{∞}{\frac{2^{2k}}{2k!} t^{2k}}[/itex]
[itex] \sum_{m=0}^{∞}{(a_{2m+2} (2m+2)(2m+1)+a_{2m+1} (2m+1)+ (1-2m) a_{2m}) t^{2m}}=\sum_{m=0}^{∞}{\frac{2^{2m}}{2m!} t^{2m}}[/itex][itex]a_{2m+2} (2m+2)(2m+1)+a_{2m+1} (2m+1)+ (1-2m) a_{2m}=\frac{2^{2m}}{2m!}[/itex]
[itex] y''+(1-t) y' + y= sin(2t)[/itex]
can someone judge this?
How could I continue it?
[itex] y=\sum_{n=0}^{∞}{a_{n} t^{n}}[/itex]
[itex] y'=\sum_{n=1}^{∞}{a_{n} n t^{n-1}}[/itex]
[itex] y''=\sum_{n=2}^{∞}{a_{n} n(n-1) t^{n-2}}[/itex]
[itex] sin(2t)=\sum_{n=0}^{∞}{\frac{2^{2n}}{2n!} t^{2n}}[/itex][itex] y''+(1-t) y' + y= sin(2t)[/itex]
[itex] \sum_{n=2}^{∞}{a_{n} n(n-1) t^{n-2}}+\sum_{n=1}^{∞}{a_{n} n (1-t)t^{n-1}}+\sum_{n=0}^{∞}{a_{n} t^{n}}=\sum_{n=0}^{∞}{\frac{2^{2n}}{2n!} t^{2n}}[/itex]
[itex] \sum_{n=2}^{∞}{a_{n} n(n-1) t^{n-2}}+\sum_{n=1}^{∞}{a_{n} n t^{n-1}}-\sum_{n=1}^{∞}{a_{n} n t^{n}}+\sum_{n=0}^{∞}{a_{n} t^{n}}=\sum_{n=0}^{∞}{\frac{2^{2n}}{2n!} t^{2n}}[/itex]
[itex] \sum_{k=0}^{∞}{a_{k+2} (k+2)(k+1) t^{k}}+\sum_{k=0}^{∞}{a_{k+1} (k+1) t^{k}}-\sum_{k=0}^{∞}{a_{k} k t^{k}}+\sum_{k=0}^{∞}{a_{k} t^{k}}=\sum_{k=0}^{∞}{\frac{2^{2k}}{2k!} t^{2k}}[/itex]
[itex] \sum_{k=0}^{∞}{(a_{k+2} (k+2)(k+1)+a_{k+1} (k+1) -a_{k} k + a_{k}) t^{k}}=\sum_{k=0}^{∞}{\frac{2^{2k}}{2k!} t^{2k}}[/itex]
[itex] \sum_{k=0}^{∞}{(a_{k+2} (k+2)(k+1)+a_{k+1} (k+1)+ (1-k) a_{k}) t^{k}}=\sum_{k=0}^{∞}{\frac{2^{2k}}{2k!} t^{2k}}[/itex]
[itex] \sum_{m=0}^{∞}{(a_{2m+2} (2m+2)(2m+1)+a_{2m+1} (2m+1)+ (1-2m) a_{2m}) t^{2m}}=\sum_{m=0}^{∞}{\frac{2^{2m}}{2m!} t^{2m}}[/itex][itex]a_{2m+2} (2m+2)(2m+1)+a_{2m+1} (2m+1)+ (1-2m) a_{2m}=\frac{2^{2m}}{2m!}[/itex]
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