Given A Second Order Linear Homogeneous Differential Equation A X Y A X Y Ao X Y 0 We Know That A Fundamental 1 (36.34 KiB) Viewed 53 times
Given A Second Order Linear Homogeneous Differential Equation A X Y A X Y Ao X Y 0 We Know That A Fundamental 2 (25.26 KiB) Viewed 53 times
Given A Second Order Linear Homogeneous Differential Equation A X Y A X Y Ao X Y 0 We Know That A Fundamental 3 (19.77 KiB) Viewed 53 times
Given a second order linear homogeneous differential equation a₂ (x)y" + a₁ (x)y + ao(x)y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions y₁, 12. 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) a₂ (x)' Then the method of reduction of order gives a second linearly independent solution as y" + p(x)y + q(x)y = 0 p(x) = y2(x) = Cy₁u = Cy₁ (x) felh q(x) = -/p(x)dx y} (x) -dx
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 1/2 = C3e²x then we can choose C = 1/3 so that y2 = ²x Given the problem y" - 2y + 10y = 0 and a solution y₁ = e* sin(3x) Applying the reduction of order method we obtain the following y₁(x) = p(x) = and e- / p(x)dx
So we have e-/ p(x)dx r(x) foto = / -dx = 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 Y2(x) = Cy₁u = So the general solution to y" - 3y + 4y = 0 can be written as y = C₁y1 + C2Y2 = C1 +0₂
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