(1) This problem will demonstrate that if you know one solution to (*) y" +p(x)y' + g(x)y=0 then you can use Abel's form

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1 This Problem Will Demonstrate That If You Know One Solution To Y P X Y G X Y 0 Then You Can Use Abel S Form 1 (43.07 KiB) Viewed 15 times

(1) This problem will demonstrate that if you know one solution to (*) y" +p(x)y' + g(x)y=0 then you can use Abel's formula to find another. Suppose that y₁ is a solution to the 2nd order linear homogeneous problem (*) and let P(t) = f p(t)dt be any antiderivative of p. (a) Show that if K is any non-zero constant and y2 satisfies 9132-₁32= Ke-P(z) then {₁,2} is linearly independent, and conclude that {31.92} is a fundamental set of solutions to (*). (b) Show that and conclude that (2) ' - = = K 31 [1 32 = 31 -P(x) yi (c) Let W = Ke-P(x), and I = J K dr. Show that V Y2 = Y/₁I K 1/₂ =1/₁1 + -P(x) y} 3₂ = ₁1 + 1/{ + W Y1 W W' -dx 1 W 3/₁ 91 7/1 and use these expressions to verify that y2 is a solution to (*). (2) Use problem 1 to find a general solution to the problem (x²2x)y" +(2-x²)y' + (2x-2)y=0; y₁ = e²