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Question

Consider the ODE system below y ′ = y² − z² ; z′ = y² + z²

Which of the following is true?

a. Solvable by trig substitution

b. Solvable using Euler method only if initial conditions are stated

c. RK 4 is not suitable for this system

d. Solvable using Euler method

Answer 1

dz/dy = (y²+z²)/(y²-z²);

dz/dy = (1+(z/y)²)/(1-(z/y)²);

Let z=ry; dz/dy = r + dr/dy*y;

dr/dy*y + r = (1 + r²)/(1 - r²);

dr/dy * y = -(r³+r²-r+1)/((r - 1)*(r + 1))

(1-r)*(r+1)/(r³+r²-r+1) dr = 1/y dy