What is a solution to the differential equation #dy/dx=sqrt(xy)sinx#?
The right hand integral can be reduced to a Fresnel integral and cannot be expressed through elementary functions.
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The solution to the differential equation ( \frac{dy}{dx} = \sqrt{xy} \sin(x) ) can be found by separating variables and integrating.

Separate variables: [ \frac{dy}{\sqrt{y}} = \sin(x) , dx ]

Integrate both sides: [ \int \frac{1}{\sqrt{y}} , dy = \int \sin(x) , dx ]
[ 2\sqrt{y} = \cos(x) + C ]
 Solve for ( y ): [ \sqrt{y} = \frac{\cos(x)}{2} + C ]
[ y = \left(\frac{\cos(x)}{2} + C\right)^2 ]
Where ( C ) is the constant of integration.
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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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