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Learn to solve first-order homogeneous ordinary differential equations using the substitution method y=vx in this comprehensive 27-minute tutorial. Master the step-by-step process of transforming non-separable homogeneous differential equations into separable form through variable substitution. Discover why the standard separation of variables method fails for certain equations and how the y=vx substitution provides an effective alternative solution approach. Follow detailed explanations of the product rule application when differentiating y=vx to obtain dy/dx = v+x(dv/dx), then learn to substitute these expressions into the original equation to create a separable differential equation. Practice identifying when equations cannot be expressed in terms of separate x-factors and y-factors, making this substitution technique essential for finding general solutions to first-order homogeneous ordinary differential equations.
