. . Substitution Method Let the differential equation be of the form dy f ax by ). (i) If a ≠ and b ¹ , then the substitution ax by z reduces the given equation to the variables separable form.
(ii) If a = or b = , then the differential equation is already in separable form. Example . Solve y ' . Given that ′ y = sin Put z = x + , so that dz = − Thus, the given equation reduces to − dz dx = sin z .
i.e., dz dx = z z . Separating the variables leads to dz z cos = dx (or) sec zdz On integration, we get tan z = x (or) tan x ) = - - Example . Solve : dy . By putting z , we have dz dx = z Hence dz z = dx .
Integrating, dz z = x Putting z = u , we have dz z udu or z z = x From which on substituting z = − , we have the general solution −− −+ = x Example . Solve: - - Given that - - Put z = x – y dz dx = – dy dx = dz - Thus, the given equation reduces to dz - = z z dz dx = – z z dz dx = z z Separating the variables, we get z z dz - - Ordinary Differential Equations z z dz