a)Find the differential equation of the family of curves y = e x (A cosx + B sinx), where A and B are arbitrary constants. Solution: - Given that, y = e x (A cosx + B sinx) ------------ (1) Differentiating (1) with respect to x, we get y ‘ = e x (-A sinx + B cosx) + e x (A cosx + B sinx) or, y ‘ = e x (-A sinx + B cosx) + y {using (1)} ------ (2) Differentiating (2) with respect to x, we get y “ = -e x (A cosx + B sinx) + e x (-A sinx + B cosx) + y’ ---- (3) Now From Equation (1) e x (-A sinx + B cosx) = y’ – y -----------(4) Hence eliminating A and B from equation (1), (3) and (4), we get y” = - y + y’ – y + y’ or, y” – 2y’ +2y = 0 ---------- (5) Equation (5) is the required differential equation. b) By eliminating the constants a and b obtain the differential equation for which xy = ae x + be -x + x 2 is a solution. Solution: - Given that, xy = ae x + be -x + x 2 ---------- (1) Differentiating (1) with respect to x,
Ordinary and Partial Differential Equations