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Question #67456

Difficulty: 8/10 | 🟢 Free

The Question

Consider the homogeneous linear system of equations in variables $u, v, w$ : $\begin{pmatrix} x & -x^2 & 0 \\ 1 & 0 & x^3 \\ y' & y & y^k \end{pmatrix} \begin{pmatrix} u \\ v \\ w \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ where $y = y(x)$ is a differentiable function, $y' = \frac{dy}{dx}$ , and $x > 0$ . The system possesses a non-trivial solution $(u, v, w) \neq (0,0,0)$ . Furthermore, the resulting differential equation for $y(x)$ is scale-invariant (i.e., the function $f(x,y)$ in $\frac{dy}{dx} = f(x,y)$ is homogeneous of degree zero). If $y(1) = 1$ , find the value of $(y(2))^{-2}$ .

Options

A.
$\frac{17}{8}$
B.
$\frac{289}{144}$
C.
$\frac{15}{8}$
D.
$\frac{17}{4}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Elementary Row Operations|Bernoulli Equation Transformation|Homogeneous Function Homogeneity

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the intersection of linear algebra and differential equations, specifically targeting the 'Homogeneous Function Homogeneity' trap.

Students must first recognize that a non-trivial solution requires the determinant of the coefficient matrix to be zero, yielding a parameterized differential equation.

The critical trap lies in the 'scale-invariant' condition: students often blindly apply Bernoulli substitutions without verifying if the resulting system satisfies the homogeneity required for the transformation to hold globally.

By enforcing $f(\lambda x, \lambda y) = f(x,y)$ , the parameter $k$ is strictly locked to 3.

Failing to verify this scale-invariance leads to applying elementary operations on a non-linear system (e.g., assuming $k=2$ ), which generates invalid integral curves and incorrect integration constants.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Elementary Row Operations
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free