Ordinary Differential Equations and Numerical Methods Review

A comprehensive set of flashcards covering key concepts and questions related to ordinary differential equations and numerical methods.

What is the order of the ODE 3y^3(d2y/dt2) – 9ty(dy/dt) = t^5(d4y/dt4) – 5ty?

Fourth

Which Runge-Kutta method orders could be used to solve the ODE in Question 1?

1st, 2nd, 3rd, and 4th order.

For the IVP dy/dt = -100*y, what is the largest step-size range for stable forward Euler?

0 < h < 0.005.

What is the backward Euler update for dy/dt = -100*y + cos(t)?

y(i+1) = y(i) + hf(t(i+1), y(i+1)).

Which integration method is valid for non-uniform x-values?

A) Trapezoidal rule.

In Python, how do you multiply matrix A and vector x?

A @ x.

If step size is reduced by half in classical RK4, global error decreases by:

1/16.

How many initial values are required for a 3rd-order ODE?

Three.

If local truncation error is O(h^4), what is the global error?

O(h^2).

For dy/dt = -100*y, what is the stability range for backward Euler?

0 < h < 0.005.

What is the term for an IVP with both fast and slow dynamics requiring very small h for stability?

Stiff.

What is the increment for the Midpoint Method?

phi = k1 + (1/2)k2.

What is required to solve a nonlinear IVP implicitly using backward Euler?

Root finding.

What is the absolute value of (y - x)^2 − x^Ty − y^Tx?

0.

For integrating a cubic, what is Simpson’s 1/3 rule truncation error compared to Simpson’s 3/8?

Less than.

Is the expression y = ax^2 + bsin(x) + c*x^3 a linear regression model?

False.

In bisection, is it correct to replace the lower bound with the midpoint if the sign test indicates it?

True.

Can linear regression always produce a curve that passes exactly through all data points?

False.

Is the ODE d2x/dt2 = ae^(xt) + b*x with x(2)=20 an IVP?

True.

Does a singular matrix system have a unique solution?

False.

Are Heun’s method and Midpoint method both RK-2 methods?

True.