How do you solve for the equation #dy/dx=(3x^2)/(e^2y)# that satisfies the initial condition #f(0)=1/2#?
First of all, I think ther is a mistake in your writing, I think you wanted to write:
This is a separable differential equations, so:
So the solution is:
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To solve the differential equation ( \frac{dy}{dx} = \frac{3x^2}{e^{2y}} ) with the initial condition ( f(0) = \frac{1}{2} ), we can separate variables and integrate both sides.

Separate variables: [ e^{2y} , dy = 3x^2 , dx ]

Integrate both sides: [ \int e^{2y} , dy = \int 3x^2 , dx ]

Solve the integrals: [ \frac{1}{2} e^{2y} = x^3 + C ] Where ( C ) is the constant of integration.

Solve for ( y ): [ e^{2y} = 2x^3 + C ] [ 2y = \ln(2x^3 + C) ] [ y = \frac{1}{2} \ln(2x^3 + C) ]

Apply the initial condition ( f(0) = \frac{1}{2} ): [ \frac{1}{2} = \frac{1}{2} \ln(C) ] [ 1 = \ln(C) ] [ C = e ]

Substitute ( C = e ) into the solution: [ y = \frac{1}{2} \ln(2x^3 + e) ]
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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