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.
You have,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.