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Section 1: Functions of Several Variables
Section 2: Limits and Continuity
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Section 3.2 – Solutions of Linear Homogeneous Equations
Section 3.2 – Solutions of Linear Homogeneous Equations
Directions.
The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
Verify that the functions \(y_1\) and \(y_2\) are solutions of the given differential equation. Do they constitute a fundamental set of solutions? \[ x^2y''-x(x+2)y'+(x+2)y=0,\qquad x>0 \quad y_1(x)=x,\quad y_2(x)=xe^x.\]
Answer
The functions \(y_1\) and \(y_2\) constitute a fundamental set of solutions. For verification that \(y_1\) and \(y_2\) are solutions of the given differential equation, see video below.
Video
To see the full video page and find related videos, click the following link.
MATH 308 WIR22A V31
If the Wronskian of \(f\) and \(g\) is \(3e^{4t}\) and \(f(t)=e^{2t}\), find \(g(t).\)
Answer
\(g(t)=(3t+ C)e^{2t}\)
Video
To see the full video page and find related videos, click the following link.
MATH 308 WIR22A V32
If the differential equation \[ty'' + 2y' + te^ty = 0\]has a fundamental set of solutions \(y_1\) and \(y_2\) and \(W(y_1, y_2)(1)\) = 2, find the value of \(W(y_1, y_2)(5).\)
Answer
\(W(y_1,y_2)(5)=\dfrac{2}{25}\)
Video
To see the full video page and find related videos, click the following link.
MATH 308 WIR22A V37