1 Point Given A Second Order Linear Homogeneous Differential Equation Z 2 Y Ai Z Y Ao Z Y 0 We Know That A Fundame 1 (49.64 KiB) Viewed 15 times
1 Point Given A Second Order Linear Homogeneous Differential Equation Z 2 Y Ai Z Y Ao Z Y 0 We Know That A Fundame 2 (30.7 KiB) Viewed 15 times
(1 point) Given a second order linear homogeneous differential equation @z(2)y" +ai(z)y +ao(z)y=0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions y/1,3/2. But there are times when only one function, call it y₁. is available and we would like to find a second linearly independent solution. We can find 3/2 using the method of reduction of order. First, under the necessary assumption the a₂ (a) 0 we rewrite the equation as y" + p(x)y' +q(a)y=0 p(x)= q(z) a₂(x)' Then the method of reduction of order gives a second linearly independent solution as 12 (2) = Cyu=Cy₁ (a) = e Sp(a)da da ao (2) a₂(x) where C is an arbitrary constant. We can choose the arbitrary constant to be anything we like. One useful choice is to choose C so that all the constants in front reduce to 1. For example, if we obtain y/2 C3e2 then we can choose = C=1/3 so that y/2 e²r
Given the problem and a solution y/₁ Applying the reduction of order method to this problem we obtain the following y} (x) = So we have p(x) = e-Jp(z)dz y} (x) =-1₁ ry" 3ry' +4y=0 dr = and e Sp(x)dr dr = Finally, after making a selection of a value for C as described above (you have to choose some nonzero numerical value) we arrive at 32(x) = Cy₁u= So the general solution to 4y" - y' + 4y = 0 can be written as y=C1y1 + 2y/2 = C1 +0₂
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