(1 point) Given a second order linear homogeneous differential equation a₂(x)y" + a₁(x)y' + a₁(x)y=0 we know that a fund

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1 Point Given A Second Order Linear Homogeneous Differential Equation A X Y A X Y A X Y 0 We Know That A Fund 1 (77.76 KiB) Viewed 22 times

(1 point) Given a second order linear homogeneous differential equation a₂(x)y" + a₁(x)y' + a₁(x)y=0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions Y₁, 92. 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 Then the method of reduction of order gives a second linearly independent solution as and a solution y₁ = x³ Applying the reduction of order method to this problem we obtain the following So we have p(x) y" + p(x)y' + q(x)y= 0 p(x) = [³ e - Sp(z)dz 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 = C3e² then we can choose C = 1/3 so that y₂ = ²x Given the problem So the general solution to 9y" -y + 4y = 0 can be written as - dx = e Y₂(x) = Cy₁u = Cy₁(x) fe y(x) y = C₁Y1 + C2Y2 a₁(x) a₂(x) x^6 C1 Sp(z)dz x²y" 5xy' +9y=0 and e-√p(z)dz y²(x) q(x) = dr = y} (x) 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 dr +6₂ ao(x) a₂(x)'