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

Answer 1

#sqrt[1 -y^2]=sqrt[1 - x^2]-1#

Grouping variables

#(y dy)/sqrt[1 - y^2]=(x dx)/sqrt[1 - x^2]#

so it is a separable differential equation.

We have also

#int (x dx)/sqrt[1 - x^2] = -sqrt[1 - x^2]# and
#int (y dy)/sqrt[1 - y^2] = -sqrt[1 -y^2]# so
#-sqrt[1 -y^2]=-sqrt[1 - x^2]+C#
and #y(0)=1# so
#-sqrt[1 -1^2]=-1+C->C = 1#

so

#sqrt[1 -y^2]=sqrt[1 - x^2]-1#
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Answer 2

To find the particular solution to the differential equation y√(1-x^2)y' - x√(1-y^2) = 0 with the initial condition y(0) = 1, we first rewrite the equation in terms of separation of variables. Then, we integrate both sides and solve for y to find the particular solution. After solving the equation, we substitute the initial condition y(0) = 1 to find the specific value of the constant of integration.

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Answer 3

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

  1. Separate variables by moving terms involving (y) to one side and terms involving (x) to the other side.
  2. Integrate both sides with respect to their respective variables.
  3. Solve for (y).
  4. Apply the initial condition (y(0) = 1) to find the particular solution.

Let's solve the equation:

[y\sqrt{1-x^2}y' = x\sqrt{1-y^2}]

[y\sqrt{1-y^2} , dy = x\sqrt{1-x^2} , dx]

[\int y\sqrt{1-y^2} , dy = \int x\sqrt{1-x^2} , dx]

After integrating both sides, we'll have:

[-\frac{1}{2}(1-y^2)^{3/2} = -\frac{1}{2}(1-x^2)^{3/2} + C]

Now, solve for (y):

[1-y^2 = (1-x^2) + C]

[y^2 = 1 - (1-x^2) - C]

[y^2 = x^2 + C]

Apply the initial condition (y(0) = 1):

[1^2 = 0 + C]

[C = 1]

So, the particular solution is:

[y^2 = x^2 + 1]

[y = \sqrt{x^2 + 1}]

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Answer from HIX Tutor

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.

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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