- #1
DivGradCurl
- 372
- 0
Problem
[tex] y^{\prime} = x^2 y [/tex]
General Comments
There must be some kind of flaw in my solution as I don't get to the same result as the one my book provides:
[tex] y = c_0 \sum _{n=0} ^{\infty} \frac{x^{3n}}{3^n n!} = c_0 e^{x^3 / 3} \qquad \fbox{CORRECT ANSWER} [/tex]
Any help is highly appreciated.
My Solution
If
[tex] y = \sum _{n=0} ^{\infty} c_n x^n \Longrightarrow x^2 y = \sum _{n=0} ^{\infty} c_n x^{n+2} = \sum _{n=3} ^{\infty} c_{n-3} x^{n-1} [/tex]
and
[tex] y^{\prime} = \sum _{n=1} ^{\infty} n c_n x^{n-1} [/tex]
Then
[tex] \sum _{n=1} ^{\infty} n c_n x^{n-1} = \sum _{n=3} ^{\infty} c_{n-3} x^{n-1} \Longrightarrow c_{n-3} = nc_n \quad n=3,4,5,\ldots [/tex]
Hence, I ultimately get
[tex] y = c_0 + c_1 x + c_2 x^2 + c_0 \sum _{n=1} ^{\infty} \left[ \frac{x^{3n}}{3\cdot 6\cdot 9\cdot \cdots \cdot \left( 3n \right) } \right] + c_1 \sum _{n=2} ^{\infty} \left[ \frac{x^{3n-2}}{4\cdot 7\cdot 10\cdot \cdots \cdot \left( 3n-2 \right) } \right] + c_2 \sum _{n=2} ^{\infty} \left[ \frac{x^{3n-1}}{5\cdot 8\cdot 11\cdot \cdots \cdot \left( 3n-1 \right) } \right] \qquad \fbox{MY ANSWER} [/tex]
Thank you very much!
[tex] y^{\prime} = x^2 y [/tex]
General Comments
There must be some kind of flaw in my solution as I don't get to the same result as the one my book provides:
[tex] y = c_0 \sum _{n=0} ^{\infty} \frac{x^{3n}}{3^n n!} = c_0 e^{x^3 / 3} \qquad \fbox{CORRECT ANSWER} [/tex]
Any help is highly appreciated.
My Solution
If
[tex] y = \sum _{n=0} ^{\infty} c_n x^n \Longrightarrow x^2 y = \sum _{n=0} ^{\infty} c_n x^{n+2} = \sum _{n=3} ^{\infty} c_{n-3} x^{n-1} [/tex]
and
[tex] y^{\prime} = \sum _{n=1} ^{\infty} n c_n x^{n-1} [/tex]
Then
[tex] \sum _{n=1} ^{\infty} n c_n x^{n-1} = \sum _{n=3} ^{\infty} c_{n-3} x^{n-1} \Longrightarrow c_{n-3} = nc_n \quad n=3,4,5,\ldots [/tex]
Hence, I ultimately get
[tex] y = c_0 + c_1 x + c_2 x^2 + c_0 \sum _{n=1} ^{\infty} \left[ \frac{x^{3n}}{3\cdot 6\cdot 9\cdot \cdots \cdot \left( 3n \right) } \right] + c_1 \sum _{n=2} ^{\infty} \left[ \frac{x^{3n-2}}{4\cdot 7\cdot 10\cdot \cdots \cdot \left( 3n-2 \right) } \right] + c_2 \sum _{n=2} ^{\infty} \left[ \frac{x^{3n-1}}{5\cdot 8\cdot 11\cdot \cdots \cdot \left( 3n-1 \right) } \right] \qquad \fbox{MY ANSWER} [/tex]
Thank you very much!