Differential Equations - maximal interval

In summary, the solution for the first problem involves using the formula f(x)=e^{bx}y_{0}+ \int{e^{b(x-t)}h(t)dt}. For the second and third problems, they can be solved using separation of variables, but it is unclear what is meant by "maximal interval."
  • #1
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Homework Statement


1. Write an interval formula for the solution
[tex]f'(x)=2f(x)+e^x[/tex]
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. [tex]f'(x)=[f(x)]^2[/tex]
f(0)=-1


Homework Equations


For
a.[tex]f'(x)=bf(x)+h(x)[/tex]
b. [tex]f(x_{0})=y_{0}[/tex]
then
[tex]e^{b(x-x_{0})}+ \int{e^{b(x-t)}h(t)dt}[/tex]

The Attempt at a Solution


1. By applying the formula, it yields
[tex]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][/tex]

2. I tried dividing f(x) from both sides of the f'(x) equation then I know that [tex]\frac{f'(x)}{f(x)}= \frac{d}{dx} \left[ lnf(x) \right] [/tex], 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 [tex] \frac{f'(x)}{f(x)}=f(x)[/tex]. Again I'm stuck. Any hints, tips, or help is greatly appreciated. Thank you.
 
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  • #2
alright, let's see.

For 1), you accidentally changed your lower bound from 0 to 1. Also, for some reason, you multiplied [tex]e^{2(x-1)}[/tex] by 0, which you should not have done.

Both 2) and 3) are simple separation of variables problem. However, I'm not entirely sure what the question means by "maximal interval". The maximum with respect to x? If so, than you don't even need to solve the differential equation. Instead just solve for x and determine whether or not its increasing or decreasing at those points.
 

1. What are differential equations?

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are used to model physical phenomena such as motion, growth, and decay.

2. What is a maximal interval for a differential equation?

A maximal interval for a differential equation is the largest interval over which the solution to the equation is valid. It is determined by the initial conditions and any constraints on the equation.

3. How is a maximal interval calculated?

The maximal interval is calculated by solving the differential equation using various techniques such as separation of variables, substitution, or using specific formulas. The resulting solution will have a domain, which is the maximal interval.

4. Why is the maximal interval important?

The maximal interval is important because it tells us the range of values for which the solution to the differential equation is valid. It also helps us determine the behavior of the solution over time and identify any critical points or singularities.

5. Can the maximal interval change?

Yes, the maximal interval can change depending on any changes in the initial conditions or constraints on the equation. It is important to always check the validity of the solution and the maximal interval when solving differential equations.

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