Question

Solve the differential equation y prime equals the product of 2 times x and the square root of the quantity 1 minus y squared .

Explain why the initial value problem y prime equals the product of 2 times x and the square root of the quantity 1 minus y squared with y(0) = 3 does not have a solution.

Answer 1

Answer:

1. y = sin(x²+C)

2. see below

Step-by-step explanation:

1. $\frac{dy}{dx} = 2x\sqrt{1-y^2}$

separation of variables

$\frac{dy}{\sqrt{1-y^2} } = 2xdx$

integration both sides$\int\frac{dy}{\sqrt{1-y^2} } = \int2xdx$

you should get :

$sin^{-1} (y) = x^2+C$ remember constant of integration!!

2. y = sin(x²+C)

3 = sin(0+C)

y(0) = 3 does not have a solution because our sin graph is not shifted vertically or multiplied by a factor whose absolute value is greater than 1, so our range is [-1,1] and 3 is not part of this range

Answer 2

The solution of the differential equation $y' = 2x \sqrt{1 -y^2$ is $y = sin (x^2 +c)$.

Differential Equation:

• A differential equation is an equation that relates variables and their derivatives.
• For eg: $\frac{dy}{dx} = f(x,y)$

How to solve the given differential equation?

• As $\frac{dy}{dx} = 2x \sqrt{1 - y^2}$, substituting y = sin(t)
  ∴ $dy = cost(t) dt$
  ∴ $cos(t) dt =2x \sqrt{1 - sin^2 (t)$
  ∴ $\int {\frac{cos(t) }{\sqrt{1 - sin^2(t)} } \, dt = \int {2x} \, dx$
  ∴$\int {\frac{cos(t) }{cos(t)} } \, dt = \int {2x} \, dx$
  ∴ $\int { } \, dt = \int {2x} \, dx$
  ∴ $t = x^2 + c$
• Substituting $t = sin^{-1} y$
  ∴$sin^{-1} y = x^2 + c$
  ∴$y = sin(x^2 + c)$
  

Thus, the solution of the given differential equation is $y = sin(x^2 + c)$

Why initial value y(0) = 3 does not have a solution for the given equation?

• Substituting x = 0 and y = 3
  ∴$3 = sin((0)^2 + c)$
  ∴ $sin(c) = 3$

As the range of sine function lies between -1 and 1, hence, the given solution for sin(c) = 3, does not exist. Thus initial value y(0) = 3 does not have a solution for the given equation.

Learn more about differential equations here:
https://brainly.com/question/1164377

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