How to you find the general solution of #dy/dx=xcosx^2#?
First we notice that
or
Hence the problem becomes
The general solution is
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To find the general solution of the differential equation ( \frac{{dy}}{{dx}} = x\cos(x^2) ), you can separate variables and integrate both sides with respect to ( x ). The steps are as follows:

Separate variables: [ \frac{{dy}}{{\cos(x^2)}} = x , dx ]

Integrate both sides: [ \int \frac{{dy}}{{\cos(x^2)}} = \int x , dx ]

Integrate the left side using a substitution. Let ( u = x^2 ), then ( du = 2x , dx ). [ \frac{1}{2} \int \frac{{dy}}{{\cos(u)}} , du = \frac{1}{2} \int x , dx ]

The integral ( \int \frac{{dy}}{{\cos(u)}} , du ) can be evaluated using a trigonometric substitution or known trigonometric identities.

Once you integrate both sides, you'll have the solution in terms of ( y ) and ( u ), which is ( x^2 ). You can then replace ( u ) with ( x^2 ) to obtain the solution in terms of ( x ).
This process will yield the general solution of the given differential equation.
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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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