The strategy to find a solution to a homogenous differential equation is to make a substitution that results in a separable equation. For a homogenous equation, either of the substitutions u = will make this work. or v = X Let u = X, or y = ux. Then dy = u dx + x du by the product rule. Use this substitution to simplify the given equation. X xy² dy + (-y³ + x³) dx = 0 xy² dy - y³ dx + x³ dx = 0 (ux)³ dx + x³ dx = 0 (u dx + x du) X y dx + u²x4 du - dx + (ux)³ dx + x³ dx = 0 du = 0 To solve the given differential equation, integrate both sides of the separated equation. We can do so as we integrate a function of y on the left side of the equation with respect to y and integrate a function of x with respect to x on the right side of the equation. = Each integration requires a simple substitution. Let u = 4y + 3 and v = 8x + 5. Then dy 1₁4 1 dy (4y + 3)² = J 1 dx (8x + 5)² 1/10/21/1/2020²0² = 2²/²/1/21/²/20 du 4 8 - 1²u - ¹ + ₁₂ = dv du 4 (8x + 5)-¹ + C₂ and dx = dv 8 )v=1 + cz 1₁8x - (4y + 3) −¹+ C₁ = To finish, replace the constant C₂ - C₁ by C to write the general solution to the given differential equation. X 0 0 5 0!

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