Seperable diff eq, differing intial-conditions

• Asphyxiated
In summary, the given equation can be rearranged to solve for y, which is equal to sin(x^2+c). For y(0)=0, the solution can be real or a multiple of pi. For y(0)=2, there are no real solutions.
Asphyxiated

Homework Statement

Solve:

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

then find a solution for:

$$y(0)=0$$

and can you find a solution for:

$$y(0)=2$$

The Attempt at a Solution

Just want to know if this is right.

First the equation can be rearranged to:

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

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

$$sin^{-1}(y) = x^{2}+c$$

$$y=sin(x^{2}+c)$$

for y(0)=0

$$0=sin(c)$$

which is valid when:

$$c = 0 \;\; or \;\; c=k \pi, \;\; \forall \; k \; \in \; Z$$

and for y(0)=2

$$2=sin(c)$$

$$c = sin^{-1}(2) = \frac {\pi}{2}-\frac {ln(4\sqrt{3}+7)}{2} i$$

which is non-real. I assume that they are looking for me to realize that the answer isn't real? I am use to dealing with complex numbers from electrical engineering so this isn't that strange to me. Although I am not sure what this really means in this context. (the context being abstract math-land.)

I think they just wanted you to realize that for sinc>1 yields no real solutions for c.

Hi Asphyxiated!

That seems very good.

Regarding the y(0)=2 case. When solve ODE's like this, it is assumed that we work in the real numbers (unless specified otherwise). So only the real solutions count. So fro y(0)=2 there are no solutions.

1. What is a separable differential equation?

A separable differential equation is a type of differential equation where the variables can be separated into two distinct functions, making it easier to solve. This type of equation is commonly used in physics and engineering to model various systems.

2. What are differing initial conditions?

Differing initial conditions refer to the different values assigned to the variables in a differential equation at the start of the problem. These initial conditions can greatly affect the solution and must be accounted for in the solving process.

3. How do you solve a separable differential equation with differing initial conditions?

To solve a separable differential equation with differing initial conditions, you must first separate the variables and integrate each side. Then, use the initial conditions to find the constants of integration and plug them back into the equation to get the final solution.

4. What are some real-world applications of separable differential equations with differing initial conditions?

There are many real-world applications of separable differential equations with differing initial conditions, such as modeling population growth, chemical reactions, and electrical circuits. These equations are also used in economics, biology, and other fields to study various systems and phenomena.

5. Are there any limitations to using separable differential equations with differing initial conditions?

While separable differential equations with differing initial conditions are useful in many cases, there are some limitations to their use. They may not accurately model complex systems or systems with changing conditions, and the solutions may not always be exact. It is important to carefully consider the problem and its assumptions before using this type of equation to ensure accurate results.

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