General solution of the differential equation ( d y d x )=a x+b y is

Step-by-step solution

Given differential equation is array ll & ( d y d x )=a x+b y & d y d x =e^ a x+b y & d y d x =e^ a x e^ b y array aligned & d y e ^ by = e ^ ax d x & e ^ - by d y- e ^ ax d x=0 & Integrating both sides, we get & e ^ - by d y- e ^ ax d x=0 & e^ -b y -b - e^ a x a +c=0 & i.e., e^ -b y b + e^ x x a =c & a e^ -b y +b e^ a x =a b c & a e^ -b y +b e^ a x =c₁ , where c₁=a b c & aligned