Given A Second Order Linear Homogeneous Differential Equation A X Y A X Y A X Y 0 We Know That A Fundamental 1 (34.92 KiB) Viewed 70 times
Given A Second Order Linear Homogeneous Differential Equation A X Y A X Y A X Y 0 We Know That A Fundamental 2 (27.29 KiB) Viewed 70 times
Given A Second Order Linear Homogeneous Differential Equation A X Y A X Y A X Y 0 We Know That A Fundamental 3 (19.74 KiB) Viewed 70 times
Given a second order linear homogeneous differential equation a₂ (x)y" + a₁ (x)y + a(x) y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions y₁, y₂. 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 y2 using the method of reduction of order. First, under the necessary assumption the a₂ (x) #0 we rewrite the equation as a₁ (x) a₂(x)' ao (x) az (x)' Then the method of reduction of order gives a second linearly independent solution as - / p(x)dx y} (x) y" + p(x)y + g(x)y = 0 p(x) = 32 (x) = Cy₁ (x) 1(x) [² q(x) = -dx
where C is an arbitrary constant. We can choose the arbitrary constant to be anything we like. Once useful choice is to choose C so that all the constants in front reduce to 1. For example, if we obtain y2 = C3e²x then we can choose C = 1/3 so that y₂ = ²x. Given the problem 25y" - 20y + 4y = 0 and a solution y₁ = e(2x/5) Applying the reduction of order method to this problem we obtain the following y} (x) = p(x) = and e- / p(x)dx =
So we have e¯ / p(x)dx y} (x) 1= -dx = S dx = 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) = So the general solution to 25y" - 20y + 4y = 0 can be written as y = C₁y1 + C2y2 = C1 +0₂
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