Help with this homework

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

Hi Henoch,

The answer to your question is option d (Solvable using the Euler Method). A little background on the Euler method: it is a numerical method that is used to approximate the solutions to ordinary differential equations (ODE's). It involves iteratively estimating the solutions by updating the values of y and z at each time step based on the current slopes of the ODE's. As time goes on, the method becomes more accurate as the time step becomes smaller, and solutions converge to the true solutions as the time step approaches zero.

Allow me to explain why the other options are incorrect. Option A is incorrect since it does not appear that this is solvable using trigonometric substitution, can be explained more later. Option B is incorrect since the Euler Method can be used regardless of the initial conditions, and the solutions will converge to the true solutions as the time step becomes smaller (much like limits and can be explained in more depth later). Option C is incorrect since the Runge-Kutta method (also known as the RK4 method) can be used to solve this problem as well, and it is a higher order method than Euler's Method which means it is likely to be more accurate than Eulers Method.