Domain of definition differential equations

[arhzz]

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.

 

[Mark44]

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.

Looks good to me. The only thing I spotted was that you wrote $\ln|\frac{300}{44}|$ in your solution. Here the absolute values aren't needed, since 300/44 is already positive.

 

[arhzz]

Looks good to me. The only thing I spotted was that you wrote $\ln|\frac{300}{44}|$ in your solution. Here the absolute values aren't needed, since 300/44 is already positive.

Aha, good to know,our professor is quite picky. Also a friend of mine pointed out that we can simplify the fraction to be 72/11. Thanks for the answer as always!