A particular solution of dy d x =(x+9 y )^2 , when x=0, y = 1 27 is

Step-by-step solution

Consider the differential equation dy d x = (x + 9y)^2 . Use the substitution v = x + 9y to transform it into a separable equation. Differentiating v with respect to x gives d v d x = 1 + 9 dy d x , from which dy d x = 1 9 ( d v d x - 1 ) . Substituting into the original equation yields 1 9 ( d v d x - 1 ) = v^2 , rearranged as d v d x = 9v^2 + 1 . This is separable: d v 9v^2 + 1 = d x . Integrate both sides: d v 9v^2 + 1 = d x . The left integral uses the standard form d u a^2 + u^2 = 1 a ( u a ) with u = 3v , a =