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?

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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?

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

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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.

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