- #1
Gale
- 684
- 2
[tex]w[f,g](t)= t^2\exp{t}\\f(t)=t[/tex]
Thats what i get, the problem is to find g(t)
So, i start; f'(t)=1
[tex]w[f,g](t)= t^2\exp{t}=f(t)g'(t)-f'(t)g(t)\\t^2\exp{t}=tg'(t)-g(t)[/tex]
divide by t,
[tex]t\exp{t}=g'(t)-\frac{g(t)}{t}[/tex]
its a 1st order linear eq. I solve for the integrating factor and get t. i multiply through and reduce
[tex](tg(t))'=t^2\exp{t}[/tex]
then i integrate with the product rule and get
[tex]tg(t)=t^2\exp{t}+2t\exp{t}+C[/tex]
divide by t and get
[tex]g(t)=t\exp{t}+2\exp{t}+\frac{C}{t}[/tex]
which is wrong. The answer in the book is
[tex}t\exp{t}+Ct[/tex]
not sure where i went wrong, i know its probably something dumb, but its late, so i need help.
~gale~
Thats what i get, the problem is to find g(t)
So, i start; f'(t)=1
[tex]w[f,g](t)= t^2\exp{t}=f(t)g'(t)-f'(t)g(t)\\t^2\exp{t}=tg'(t)-g(t)[/tex]
divide by t,
[tex]t\exp{t}=g'(t)-\frac{g(t)}{t}[/tex]
its a 1st order linear eq. I solve for the integrating factor and get t. i multiply through and reduce
[tex](tg(t))'=t^2\exp{t}[/tex]
then i integrate with the product rule and get
[tex]tg(t)=t^2\exp{t}+2t\exp{t}+C[/tex]
divide by t and get
[tex]g(t)=t\exp{t}+2\exp{t}+\frac{C}{t}[/tex]
which is wrong. The answer in the book is
[tex}t\exp{t}+Ct[/tex]
not sure where i went wrong, i know its probably something dumb, but its late, so i need help.
~gale~