Consider the following functions. f₁(x) = ex, f₂(x) = ex, f3(x) = sinh(x) g(x) = C₁f1(x) + C₂2(x) + C33(x) Solve for C₁, C₂, and c3 so that g(x) = 0 on the interval (-∞, ∞). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution (0, 0, 0).) (C₁, C₂, C3) = -{[ Determine whether f1, f2, f3 are linearly independent on the interval (-∞, ∞). O linearly dependent O linearly independent.
Consider the following functions. f₁(x) = ex, f₂(x) = ex, f3(x) = sinh(x) g(x) = C₁f1(x) + C₂2(x) + C33(x) Solve for C₁, C₂, and c3 so that g(x) = 0 on the interval (-∞, ∞). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution (0, 0, 0).) (C₁, C₂, C3) = -{[ Determine whether f1, f2, f3 are linearly independent on the interval (-∞, ∞). O linearly dependent O linearly independent.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the following functions.
f₁(x) = ex, f₂(x) = ex, f3(x) = sinh(x)
g(x) = C₁f1(x) + C₂f₂(x) + C33(x)
Solve for C₁, C₂, and c3 so that g(x) = 0 on the interval (-∞, ∞). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution (0, 0, 0).)
(C₁, C₂, C3) =
Determine whether f₁, f2, f3 are linearly independent on the interval (-∞, ∞0).
O linearly dependent
O linearly independent
Consider the differential equation
x²y" - 9xy' + 24y = 0; x4, x6, (0, ∞).
Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval.
The functions satisfy the differential equation and are linearly independent since W(x¹, x) =
0 for 0 < x < 00.
Form the general solution.
y =
Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.
2x2y" + 5xy' + y = x² = x;
y = ₁x¹/² + ₁₂x¹ + x² -1x, (0, 0)
15
6
The functions x-1/2 and x-1 satisfy the differential equation and are linearly independent since W(x-1/2, x-¹)=
form a fundamental set of solutions of the associated homogeneous equation, and y=
#0 for 0 < x <∞, So the functions x-1/2 and x-1
is a particular solution of the nonhomogeneous equation.
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