# 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.
1. Verify that the functions $$y_1$$ and $$y_2$$ are solutions of the given differential equation. Do they constitute a fundamental set of solutions? $x^2y''-x(x+2)y'+(x+2)y=0,\qquad x>0 \quad y_1(x)=x,\quad y_2(x)=xe^x.$

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.

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2. If the Wronskian of $$f$$ and $$g$$ is $$3e^{4t}$$ and $$f(t)=e^{2t}$$, find $$g(t).$$

$$g(t)=(3t+ C)e^{2t}$$

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3. 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 $$W(y_1, y_2)(5).$$

$$W(y_1,y_2)(5)=\dfrac{2}{25}$$

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