- #1

arhzz

- 268

- 52

- Homework Statement
- Check for the biggest domain of definition

- Relevant Equations
- DE

Hello.

Considering this DE;

$$ x^7 x' = (x^8-300)t^6 $$ with inital value x(0) = -2

Now the solution for the initial value should be

C = -44;

And for x(t) I get ;

$$x(t) = (-44 e^{\frac{8}{7} t^7} + 300)^{\frac{1}{8}}$$

Now to get the biggest domain of definition I did this;

$$ -44 e^{\frac{8}{7} t^7} + 300 > 0 $$

And tried to isolate t; Here is how the manipulation went

$$ -44 e^{\frac{8}{7} t^7} > -300 $$ Now I divided with -44 and since I am dividing with a negative number the > sign should turn into <.

$$ e^{\frac{8}{7} t^7} < \frac{300}{44} $$ ln on both sides

$$ \frac{8}{7} t^7 < ln |\frac{300}{44}| $$ now dividing with 8/7 (I wrote it as multiplication with 7/8) and the 7th root I wrote as a fraction 1/7,hence I get;

$$ t < ( \frac{7}{8} ln |\frac{300}{44}| )^\frac{1}{7} $$

Is my solution correct? Am I allowed to solve these types of problems this way.

Considering this DE;

$$ x^7 x' = (x^8-300)t^6 $$ with inital value x(0) = -2

Now the solution for the initial value should be

C = -44;

And for x(t) I get ;

$$x(t) = (-44 e^{\frac{8}{7} t^7} + 300)^{\frac{1}{8}}$$

Now to get the biggest domain of definition I did this;

$$ -44 e^{\frac{8}{7} t^7} + 300 > 0 $$

And tried to isolate t; Here is how the manipulation went

$$ -44 e^{\frac{8}{7} t^7} > -300 $$ Now I divided with -44 and since I am dividing with a negative number the > sign should turn into <.

$$ e^{\frac{8}{7} t^7} < \frac{300}{44} $$ ln on both sides

$$ \frac{8}{7} t^7 < ln |\frac{300}{44}| $$ now dividing with 8/7 (I wrote it as multiplication with 7/8) and the 7th root I wrote as a fraction 1/7,hence I get;

$$ t < ( \frac{7}{8} ln |\frac{300}{44}| )^\frac{1}{7} $$

Is my solution correct? Am I allowed to solve these types of problems this way.

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