(iv) by reducing to differential equation ex. solve ux)= 1 - 2x - 4x +13+6(x-t)-4(x -tju(t) dt. sol. by using leibnitz rule differentiating both sides of given

(IV) By Reducing to Differential Equation Ex. Solve ux)= 1 - 2x - 4x +13+6(x-t)-4(x -tju(t) dt. Sol. By using Leibnitz rule differentiating both sides of given equation wrt x, we get u'(x)= -2 - 8x +6-8(x-t)Ju(t) dt +3u(x) u'x) =-8 - 8u(t) dt+6u(x) + 3u' (x) u" (x) = -8u(x) + 6u"(x) + 3u" .. ( and u"x) - 9u'"(x) + 8u(x) = 0 which general solution is ux)=aex + bex +ce-2x u'(x) = ae' + 4be 2ce-x u"(x) = aex + 16be + 4ce-2x Since, u(0) = 1 (From given integral equation) u'(0) =2+ 3u(0) = -2 + 3 = 1 u(0) = -8 + 6u(0) + 3u'(0) = -8+6+ 3 = 1 [From Eq.(0 Now, From Eq.(ii), (iv) and (v) [From Eq.0 a +b+ C = 0, a+ 4b - 2c = 0, a + 16b + 4c 0 a = 1, b = 0, c = 0 From Eq.(ii), required solution is u(x) = ex " plz explain this problem i can not understand