Consider the differential equation

dy/dx= 3x^2/e^2y

a) Find a solution y=f(x) to the differential equation satisfying f(0)=1/2

b) Find the domain and range of the function f found in part a.

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- Apr 3rd 2006, 04:41 PMfrozenflamesCalculus Help
Consider the differential equation

dy/dx= 3x^2/e^2y

a) Find a solution y=f(x) to the differential equation satisfying f(0)=1/2

b) Find the domain and range of the function f found in part a. - Apr 3rd 2006, 07:26 PMThePerfectHackerQuote:

Originally Posted by**frozenflames**

This, is a 'seperable differencial equation' .

Thus, express as,

Thus, integrating,

.

Thus,

Thus,

Thus,

Thus,

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Now, we have that

Thus,

Thus,

Meaning, the function is,

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The domain of

is what values can take. Notice the logarithm, which is only defined for positive real numbers. Thus,

Solving this inequality, and rationalizing the root

.

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The range for

is the values or such as, this equation is a solution for . Notice, that this equation has a solution for any thus, the range is all real numbers.