Paper Wings
Jul21-08, 02:19 PM
1. The problem statement, all variables and given/known data
1. Write an interval formula for the solution
f'(x)=2f(x)+e^x
f(1)=0
Explicitly find the maximal interval I about 0 on which we can solve the following differential equations
2. f'(x) = xf(x)
f(0)=1
3. f'(x)=[f(x)]^2
f(0)=-1
2. Relevant equations
For
a.f'(x)=bf(x)+h(x)
b. f(x_{0})=y_{0}
then
e^{b(x-x_{0})}+ \int{e^{b(x-t)}h(t)dt}
3. The attempt at a solution
1. By applying the formula, it yields
f(x)=e^{2(x-1)}0+ \int_0^x{e^{2(x-t)}dt}=\left[ -\frac{1}{3}e^{3x-2t}\right]\right|^{x}_{1}= \left[ -\frac{1}{3}e^x- \left(- \frac{1}{3}e^{3x-2} \right) \right]
2. I tried dividing f(x) from both sides of the f'(x) equation then I know that \frac{f'(x)}{f(x)}= \frac{d}{dx} \left[ lnf(x) \right] , but I do not know how to procede from here.
3. I've tried using the same method with part 2 except that I divide one f(x) from both sides which yields \frac{f'(x)}{f(x)}=f(x). Again I'm stuck. Any hints, tips, or help is greatly appreciated. Thank you.
1. Write an interval formula for the solution
f'(x)=2f(x)+e^x
f(1)=0
Explicitly find the maximal interval I about 0 on which we can solve the following differential equations
2. f'(x) = xf(x)
f(0)=1
3. f'(x)=[f(x)]^2
f(0)=-1
2. Relevant equations
For
a.f'(x)=bf(x)+h(x)
b. f(x_{0})=y_{0}
then
e^{b(x-x_{0})}+ \int{e^{b(x-t)}h(t)dt}
3. The attempt at a solution
1. By applying the formula, it yields
f(x)=e^{2(x-1)}0+ \int_0^x{e^{2(x-t)}dt}=\left[ -\frac{1}{3}e^{3x-2t}\right]\right|^{x}_{1}= \left[ -\frac{1}{3}e^x- \left(- \frac{1}{3}e^{3x-2} \right) \right]
2. I tried dividing f(x) from both sides of the f'(x) equation then I know that \frac{f'(x)}{f(x)}= \frac{d}{dx} \left[ lnf(x) \right] , but I do not know how to procede from here.
3. I've tried using the same method with part 2 except that I divide one f(x) from both sides which yields \frac{f'(x)}{f(x)}=f(x). Again I'm stuck. Any hints, tips, or help is greatly appreciated. Thank you.