Differential Equations

Transformation of some equation in the form in which variables are separable . Equation of the form Dy/dx = ƒ (ax + by + c) or, dy/dx = ƒ (ax + by) can be reduced to an equation in which variables can be separated. For this purpose we use the substitution ax + by + c = v or, ax + by = v Example: Solve: dy/dx = (4x + y +1) 2 Given that, dy/dx = (4x + y +1) 2 ----------- (1) Let, 4x + y +1 = v -------------- (2) Differentiating equation (1) with respect to x, we get 4 + dy/dx = dv/dx or, dy/dx = dv/dx – 4 --------------- (3) We get from equation (2) & (3), dv/dx – 4 = v 2 or, dv/dx = 4 + v 2 or, dx = dv/(4 + v 2 ) Integrating, x + c = (1/2) tan -1 (v/2) or, 2x + 2c = tan -1 (v/2) or, v = 2 tan (2x + 2c) or, 4x + y +1 = 2 tan (2x + 2c) Answer: Example: Solve: dy/dx = sin (x + y) + cos (x + y) Given that, dy/dx = sin (x + y) + cos (x + y) ------------- (1) Let, x + y = v ------------(2) Differentiating equation (1) with