# How do you find the particular solution to #ysqrt(1-x^2)y'-x(1+y^2)=0# that satisfies y(0)=sqrt3?

Use the separation of variables method, integrate both sides, and then use the specified point to evaluate the constant.

Move the second term to the right side:

Integrate both sides:

Multiply both sides by 2:

Use the exponential function:

Adding a constant in the exponent is the same as multiplying by a constant:

Subtract 1 from both sides:

Square root both sides:

Square both sides and solve for C:

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To find the particular solution to the differential equation ( y\sqrt{1-x^2}y' - x(1+y^2) = 0 ) that satisfies ( y(0) = \sqrt{3} ), follow these steps:

- Rewrite the differential equation in standard form.
- Separate variables and integrate both sides.
- Solve for ( y ) to find the particular solution.
- Use the initial condition ( y(0) = \sqrt{3} ) to determine the constant of integration.

Step 1: Rewrite the differential equation: [ y\sqrt{1-x^2} \frac{dy}{dx} = x(1+y^2) ]

Step 2: Separate variables and integrate: [ \int \frac{y}{1+y^2} , dy = \int \frac{x}{\sqrt{1-x^2}} , dx ]

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

[ \ln(1+y^2) = -\arcsin(x^2) + C_1 ]

[ 1 + y^2 = e^{-\arcsin(x^2) + C_1} ]

[ y^2 = e^{-\arcsin(x^2) + C_1} - 1 ]

[ y = \pm \sqrt{e^{-\arcsin(x^2) + C_1} - 1} ]

Step 4: Apply the initial condition ( y(0) = \sqrt{3} ) to find ( C_1 ): [ \sqrt{3} = \sqrt{e^{-\arcsin(0)} - 1} ] [ \sqrt{3} = \sqrt{e^0 - 1} ] [ \sqrt{3} = \sqrt{1} ] [ C_1 = 0 ]

Therefore, the particular solution satisfying ( y(0) = \sqrt{3} ) is: [ y = \sqrt{e^{-\arcsin(x^2)}} ]

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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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