# What is a solution to the differential equation #sqrtx+sqrtydy/dx# with y(1)=4?

To find its General Soln. , we integrate term-wise , i.e.,

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The given differential equation is:

[ \sqrt{x} + \sqrt{y} \frac{dy}{dx} ]

Given the initial condition ( y(1) = 4 ), we can solve this equation by separating variables and then integrating both sides.

- Separate variables:

[ \frac{1}{\sqrt{y}} , dy = -\frac{1}{\sqrt{x}} , dx ]

- Integrate both sides:

[ \int \frac{1}{\sqrt{y}} , dy = \int -\frac{1}{\sqrt{x}} , dx ]

[ 2\sqrt{y} = -2\sqrt{x} + C ]

- Apply the initial condition ( y(1) = 4 ) to find the constant ( C ):

[ 2\sqrt{4} = -2\sqrt{1} + C ]

[ 4 = -2 + C ]

[ C = 6 ]

- Substitute ( C ) back into the equation:

[ 2\sqrt{y} = -2\sqrt{x} + 6 ]

- Solve for ( y ):

[ \sqrt{y} = -\sqrt{x} + 3 ]

[ y = (-\sqrt{x} + 3)^2 ]

[ y = x - 6\sqrt{x} + 9 ]

So, the solution to the differential equation ( \sqrt{x} + \sqrt{y} \frac{dy}{dx} ) with the initial condition ( y(1) = 4 ) is ( y = x - 6\sqrt{x} + 9 ).

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When evaluating a one-sided 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 one-sided 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 one-sided 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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