Froggy Jumps
Core topics in linear algebra, graphs, and ODEs
1
To find the particular solution of y'' - 3y' + 2y = e^(3x) using the method of Undetermined Coefficients, what is the appropriate guess for the particular solution Yp?
2
Which of the following differential equations is a second-order linear differential equation with variable coefficients?
3
For a second-order linear homogeneous ODE with complex conjugate roots λ = α ± iβ from its characteristic equation, what is the general solution?
4
The method of Variation of Parameters is suitable for finding particular solutions for non-homogeneous linear differential equations with:
5
What is the general form of a homogeneous Cauchy-Euler equation?
6
The method of Undetermined Coefficients is used to find a particular solution for which type of differential equation?
7
If the characteristic equation of ay'' + by' + cy = 0 has a repeated real root r, what is the form of the general solution?
8
For a second-order linear homogeneous differential equation with constant coefficients ay'' + by' + cy = 0, if the characteristic equation has two distinct real roots r1 and r2, what is the general solution?
9
Which method is typically used to solve a first-order linear ODE like dy/dx + y/x = x?
10
Solve the separable differential equation dy/dx = 2x/y, with y(0)=1.
11
Which type of first-order ODE can be written in the form M(x, y)dx + N(x, y)dy = 0, where ∂M/∂y = ∂N/∂x?
12
A Bernoulli's differential equation is of the form dy/dx + P(x)y = Q(x)y^n. What substitution transforms it into a linear equation?
13
For the first-order linear differential equation dy/dx + P(x)y = Q(x), what is the integrating factor?
14
Which of the following is an example of an exact first-order ordinary differential equation?
15
Which of the following describes an 'incidence matrix' of a graph?
16
A 'circuit' in a graph is defined as:
17
For a graph with n vertices, what does an adjacency matrix A represent?
18
Two graphs G1 and G2 are isomorphic if:
20
A graph in which every pair of distinct vertices is connected by a unique edge is called a:
21
Which type of graph allows multiple edges between the same pair of vertices but no loops?
22
What is a 'simple graph' in graph theory?
23
The matrix A = [[cosθ, -sinθ], [sinθ, cosθ]] represents a linear transformation. What type of transformation is it?
24
Consider the transformation T(x, y) = (x + y, x - y). Is T a linear transformation?
25
If a linear transformation T: R^2 -> R^2 projects every vector onto the x-axis, what is its standard matrix?
26
Which of the following is a common application of linear transformations in image processing?
27
What is the standard matrix for a linear transformation T: R^2 -> R^2 that reflects vectors across the x-axis?
28
A transformation that rotates a vector in a 2D plane by a fixed angle θ is an example of which type of linear transformation?
29
Which of the following is the defining property of a linear transformation T: V -> W?
30
Which of the following differential equations is not a second-order linear differential equation with constant coefficients?
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