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Bounceback
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Solve [itex]f'(x)*a(x)+a'(x)=1[/itex] For a(x)
(there's another less important question at the bottom)
Background behind equation (trying to find a function to integrate any e^f(x)):
[itex]\int e^{f(x)}\,dx=e^{f(x)}*a(x)[/itex]
[itex]e^{f(x)}=e^{f(x)}*{f'(x)}*a(x)+e^{f(x)}*a'(x)[/itex]
[itex]1=f'(x)*a(x)+a'(x)[/itex]A few of my attempts:
[itex]df(x)*a(x)+da(x)=d(x)[/itex]
[itex]\int a(x)\,df(x)+a(x)=x[/itex]
[itex]a(x)=x-\int a(x)\,df(x)[/itex]
[itex]a(x)=x-\int x-\int x-\int x ...\,df(x)\,df(x)\,df(x)[/itex]
[itex]a(x)=x-\int x\,df(x)+\iint x\,d^2f(x)-\iiint x\,d^3f(x)...[/itex]
Note, this attempt only works if f(x) is an integrable function
This method doesn't work however, since [itex]\int e^x\,dx=e^x[/itex], You know the value of a and f(x) (1 and x respectively).
Substituting into the equation, the method stops working at line 2. Assuming that this is because you can only integrate a side of an equation with respect to a single function, I tried a different method.
[itex]f'(x)*a(x)=1-a'(x)[/itex]
[itex]\int f'(x)*a(x)\,dx=\int 1-a'(x)\,dx[/itex]
[itex]\int f'(x)*a(x)\,dx=x-a(x)[/itex]
A rearranged version of line two...
Where have I gone wrong in my attempts?
(there's another less important question at the bottom)
Background behind equation (trying to find a function to integrate any e^f(x)):
[itex]\int e^{f(x)}\,dx=e^{f(x)}*a(x)[/itex]
[itex]e^{f(x)}=e^{f(x)}*{f'(x)}*a(x)+e^{f(x)}*a'(x)[/itex]
[itex]1=f'(x)*a(x)+a'(x)[/itex]A few of my attempts:
[itex]df(x)*a(x)+da(x)=d(x)[/itex]
[itex]\int a(x)\,df(x)+a(x)=x[/itex]
[itex]a(x)=x-\int a(x)\,df(x)[/itex]
[itex]a(x)=x-\int x-\int x-\int x ...\,df(x)\,df(x)\,df(x)[/itex]
[itex]a(x)=x-\int x\,df(x)+\iint x\,d^2f(x)-\iiint x\,d^3f(x)...[/itex]
Note, this attempt only works if f(x) is an integrable function
This method doesn't work however, since [itex]\int e^x\,dx=e^x[/itex], You know the value of a and f(x) (1 and x respectively).
Substituting into the equation, the method stops working at line 2. Assuming that this is because you can only integrate a side of an equation with respect to a single function, I tried a different method.
[itex]f'(x)*a(x)=1-a'(x)[/itex]
[itex]\int f'(x)*a(x)\,dx=\int 1-a'(x)\,dx[/itex]
[itex]\int f'(x)*a(x)\,dx=x-a(x)[/itex]
A rearranged version of line two...
Where have I gone wrong in my attempts?
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