Question

Find the solution of the system x²=4y,y²=2x, where primes indicate derivatives with respect to t, that satisfies the initial condition x(0)=2,y(0)=2.

x = ____
y = ____
Based on the general solution from which you obtained your particular solution, complete the following two statements: The critical point (0,0) is
A. asymptotically stable
B. stable
C. unstable
and is a
A. node
B. center
C. saddle point
D. spiral

Answer 1

The solution to the system x²=4y,y²=2x that satisfies the initial condition x(0)=2,y(0)=2 is x=8 and y=±4. The critical point (0,0) is unstable (option C) and is a saddle point (option C).

Step-by-step explanation:

To find the solution of the system x²=4y,y²=2x, we can substitute one equation into the other.

Let's substitute y²=2x into x²=4y:

x²=4(2x)

Simplifying this equation, we get:

x²=8x

Dividing both sides by x, we get:

x=8

Now, substitute this value of x into y²=2x:

y²=2(8)

Simplifying, we get:

y²=16

Taking the square root of both sides, we get:

y=±4

So, the solution to the system x²=4y,y²=2x that satisfies the initial condition x(0)=2,y(0)=2 is x=8 and y=±4. The critical point (0,0) is unstable and is a saddle point.