1. Find the general solution to y" - y = 0. 2. Take y" -2xy' + 4y = 0. (a) Show that y = 1-2r2 is a solution. (b) Use reduction of order to find a second linearly independent solution. (c) Write down the general solution. 3. Find the solution of y" - 17y' + 30y= 0 with y(0) = 1, y'(0) = −11. 4. Find the general solution to y" - 12y' + 36y= 0.

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Are the functions y₁(x) = e, y2(x) = ze a fundamental set of solutions for the equation y" - 2y + y = 0? Justify your answer. If yes, write the general solution of y" - 2y + y = 0.

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x?y" + 2xy' – 2y = 0; x > 0; y = y(x) (1) 3. Determine if the two solutions in Questions 1 and 2 are linearly independent.

7a please show all steps for only a

For the equation ay" + by' + cy = 0, if b² – 4ac = 0 (the repeated root case where the root is r = -b/2a) we learned that one solution is Y1 = e-bt/2a, and the second linearly independent solution is y, = te-bt/2a We were told that the second solution is found via the Reduction of Order method. Apply the Reduction of Order method to the equation using the first solution (Do the calculations, this will take about 5 or so minutes) Which of the following equations does it reduces to? au" + 2bu' - u + cu = 0 2a 62 àu" – 2bu' +u + cu = 0 au" = 0 au" + 2bu' + Eu + cu = 0 O O O

Find two linearly independent solutions of 2x2y"-xy'+(-6x+1)y=0, x>0 of the form

(a) Form the difference equation whose solution is yk = ci + (c2 + c3k)(-1)*, where c1, c2 and cs are arbitrary constant.

Eliminate C from the equation y=3x²+ Ce-2x (A) y'-2y-6x²+6x=0 B y'- 2y + 6x² - 6x=0 Ⓒ2y+y'+3x²+ 6x=0 D2y+y¹-6x² - 6x=0