How do you find the general solution to #dy/dx+e^(x+y)=0#?
this is separable
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To find the general solution to the differential equation ( \frac{dy}{dx} + e^{x+y} = 0 ), first separate the variables by moving terms involving ( y ) to the left side and terms involving ( x ) to the right side. Then integrate both sides with respect to their respective variables. The solution will involve an arbitrary constant ( C ). The steps are as follows:
 Rearrange the equation to separate variables:
[ \frac{dy}{dx} = e^{x+y} ]
 Integrate both sides:
[ \int dy = \int e^{x+y} dx ]
 Integrate with respect to ( y ) on the left side and with respect to ( x ) on the right side:
[ y = \int e^{x+y} dx + C ]

Integrate ( e^{x+y} ) with respect to ( x ). This might require using techniques such as substitution or integration by parts.

After integrating, solve for ( y ) to obtain the general solution.
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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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