Given A Second Order Linear Homogeneous Differential Equation A X Y A X Y Ao X Y 0 We Know That A Fundamental 1 (37.03 KiB) Viewed 49 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 (25.27 KiB) Viewed 49 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 (20.08 KiB) Viewed 49 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, V2. 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) = q(x) = -/p(x)dx 12(x) = Cy₁ = Cy Cy₁ (x) (x) [e- / M 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 y2 = C3e²x then we can choose C = 1/3 so that y₂ = ²x Given the problem x²y" + 5xy - 12y = 0 and a solution y₁ = x² 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 -/p(x)dx y(x) e) -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 25y" - 5y + 4y = 0) can be written as y = C₁y₁ + C₂y2 = C₁ +0₂
Join a community of subject matter experts. Register for FREE to view solutions, replies, and use search function. Request answer by replying!