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Comparing the Efficiency of the Bernoulli Method and the Homogeneous (u = y/x) Method for Solving First-Order Differential Equations Authors: Joyce Lu, Hilary Bayer Math 2A | Winter 2023 | Foothill College OVERVIEW IVP & METHODS We begin with an equation that is both Bernoulli and homogeneous (u = ). Second, we apply the Bernoulli method. We examine the efficiency of the Bernoulli method and the homogeneous method for solving first-order differential dy = 2 + 7-, with initial values x = 1, y(x ) = 2. (Note x # 0.) 1 dy v dx = 7 rewrite in proper form equations. Determining the efficiency of methods of solving differential equations is useful for computational First, we apply the homogeneous method. du = 1 dy identify substitution to make linear applications as it enables the design of an efficient u = -, y = ux, dx dy - = u + x dx substitution for homogeneous method algorithm. In turn, efficient computer algorithms open up du = u+ 7u the possibility for programming applications of differential by substitution dx +4= = by substitution equations. 1 du = S+dx by separation of variables J-dx A first-order differential equation that can be solved using u(x) = e =X find integrating factor for linear ODE either the Bernoulli method or the homogeneous method ("Bernoulli-homogeneous equation") is of the form 14 - 17 du = S-dx by partial fraction decomposition x du + 2vx = - 7 multiply ODE by integrating factor dy + P(x)y = Q(x)y Inlul - In|7 + 11 = In/x| + c (x V )' = -7 by product rule where dx P ( x ) = k x , Q ( x ) = k x , k. k r ER Inlul - In|7u+1= e Inlx|+ , S (x v)'dx = ]- 7dx To determine the efficiency of each respective method, we 7u+1 = XC solve a given Bernoulli-homogeneous equation with both x v = - 7v+ C2 the Bernoulli method and the homogeneous technique. The 7+1 = XC invert substitution * = =+C2 invert substitution number of steps necessary to obtain the final result are then compared between methods. 1 - 7xc y C , - 7 REFERENCES RESULTS SUMMARY Homogeneous Differential Equations. Math is Fun. These solutions can be reconciled Homogeneous: 5 steps (substitute, separate https://www.mathsisfun.com/calculus/differential-equation y = 1- 7x6 variables, integrate, invert substitution, solve for y) s-homogeneous.html. Bernoulli: 5 steps (substitute, find integrating Bernoulli Differential Equations. Paul's Online Notes. Finally, we use the initial values factor, rewrite ODE with integrating factor, invert https://tutorial.math.lamar.edu/classes/de/bernoulli.aspx * = 1, y(x ) = 2 to find c,. substitution, solve for y) Linear Differential Equations. Paul's Online Notes. 2 = 7-71 6, = 7.5 Although both techniques share the same https://tutorial.math.lamar.edu/classes/de/Linear.aspx number of steps, note that the process for Ryker, instructor, appointment & office hours homogeneous substitution is more The solution to the IVP is y = 75 -7x- Fig. 1. Graphically reconciling solutions. difficult than Bernoulli