Question

Differential Equations: Check each answer to prove the answer is correct.

Solve the equation (x+y)dx + xdy = 0 by two different methods: a) by substitution (u = y/x) b) as an exact differential equation. c) show your two answers are equivalent (i.e. differ by a constant)

Answer 1

(x + y) dx + x dy = 0 A differential equation is said to be exact of form like M dx + N dy = 0 if Partialdifferential M/Partialdifferential Y = Partialdifferential N/Partialdifferential x here M = (x + y), N = x then Partialdifferential/Partialdifferential y (x + y) = Partialdifferential/Partialdifferential x (x) 1 = 1 true Now the solution of this type of differential equation is rightarrow integral M dx + integral (terms of N not contain (x)) dy Then integral (x + y) dx + integral [0] dy = c x^2/2 + yx = c x^2 + 2yx = 2c 2tx = 2c - x^2 y = 2c - x^2 /2x \r\n (x + y) dx + x dy = 0 x dy = -(x + y) dx dy/dx = -(x + y/x) = -(1 + y/x) put y = ux = > u = Y/x dy/dx = u + x du/dx