Jen_Jer_888
New member
- Joined
- Jul 28, 2011
- Messages
- 1
The problem:
Solve the fluxional equation \(\displaystyle \frac{\dot{y}}{\dot{x}} =\frac{ 2}{x} + 3 - x^{2}\) by first replacing x by (x + 1) and then using power series techniques.
My feeble attempt at a solution:
First, I believe the fluxional (y with a dot on top)/(x with a dot on top) was just Newton's language and notation for the derivative dy/dx, so I rewrote the equation as \(\displaystyle \frac{dy}{dx} = \frac{2}{x} + 3 - x^{2}\). Then I replaced x by (x + 1) like it says, getting: \(\displaystyle \frac{dy}{d(x+1)} = \frac{2}{(x+1)} + 3 - (x+1)^{2}\)
From there, I attempted to set it equal to the sigma series for the derivative of a power series, so: \(\displaystyle \frac{2}{(x+1)} + 3 - (x+1)^{2} = \sum_{n=1}^{\infty}n\cdot a_{n}\cdot (x+1)^{n-1} = a_{1} + 2\cdot a_{2}*(x+1)+ 3*a_{3}*(x+1)^{2} + 4*a_{4}*(x+1)^{3} + ....\)
I don't know where to go from there. I'm not even sure how the substitution helps. Since there is no y or higher order derivative, I see no basis to compare series coefficients.
Any help would be appreciated!
Solve the fluxional equation \(\displaystyle \frac{\dot{y}}{\dot{x}} =\frac{ 2}{x} + 3 - x^{2}\) by first replacing x by (x + 1) and then using power series techniques.
My feeble attempt at a solution:
First, I believe the fluxional (y with a dot on top)/(x with a dot on top) was just Newton's language and notation for the derivative dy/dx, so I rewrote the equation as \(\displaystyle \frac{dy}{dx} = \frac{2}{x} + 3 - x^{2}\). Then I replaced x by (x + 1) like it says, getting: \(\displaystyle \frac{dy}{d(x+1)} = \frac{2}{(x+1)} + 3 - (x+1)^{2}\)
From there, I attempted to set it equal to the sigma series for the derivative of a power series, so: \(\displaystyle \frac{2}{(x+1)} + 3 - (x+1)^{2} = \sum_{n=1}^{\infty}n\cdot a_{n}\cdot (x+1)^{n-1} = a_{1} + 2\cdot a_{2}*(x+1)+ 3*a_{3}*(x+1)^{2} + 4*a_{4}*(x+1)^{3} + ....\)
I don't know where to go from there. I'm not even sure how the substitution helps. Since there is no y or higher order derivative, I see no basis to compare series coefficients.
Any help would be appreciated!
