Question

Consider the homogeneous linear 2nd order DE: sin(x) y'' + ln(x) y' + y = 0. Suppose y1 and y2 are solutions.
a. Show that y1+y2 is also a solution.
b. Show that αy1 is a solution for any real number α
c. What does this say about the set of solutions?
d. Is the same true for the non-homogeneous DE sin(x) y'' + ln(x) y' + y = 1?

Answer 1

Where are you stuck? I can help you figure it out.

Answer 2

I need help on a,b,c and d.
for part I started like this:
\[\sin(x)y_1'' +\ln(x)y_1'+y_1=0\]
\[\sin(x)y_2''+\ln(x)y_2'+y_2=0\]
\[\sin(x)y_1''+\sin(x)y_2'' +\ln(x)y_1'+\ln(x)y_2'+y_1+y_2=0\]
\[\sin(x)(y_1+y_2)''+\ln(x)(y_1+y_2)'+(y_1+y_2)=0\]

Answer 3

That's correct for the first part!

Answer 4

b.\[
\begin{align*}
&\phantom{{}={}}\sin(x) (ay_1'') + \ln(x) (ay_1') + ay_1\\
&=a\left(\sin(x) y_1'' + \ln(x) y_1' + y_1\right)\\
&=a(0)\\
&=0
\end{align*}
\]
c. Look at the axioms of a vector space.
d. Try it out yourself.