Modern Differential Equations Reduction of Order can be appl

Modern Differential Equations

Reduction of Order can be applied to 3rd order ODEs, similarly to how we applied it to 2nd order ODEs. Consider the 3rd order ODE: xy\"\' - xy\" + y\' - y =0 Given that y_1 = e^x is a solution to the above ODE, use y_2 = vy_1 to reduce this equation to a second order equation by the change of variables w = y\'. State the solution, plot it, and determine the end behavior.

Solution

e x ydx = (e 2x + 1)dy y(0) = 1 Solution: Z e x e 2x + 1 dx = Z dy y Using the substitution u = e x , du = e xdx on the integral on the left Z du u 2 + 1 = Z dy y arctan(u) + C = ln |y| arctan(e x ) + C = ln |y| Applying our initial conditions arctan(1) + C = ln(1) 4 + C = 0 C = 4 arctan(e x ) 4 = ln |y| Solving for y gives the solution to the differential equation: y = e arctan(e x )