Question

A particular solution and a fundamental solution set are given for the nonhomgeneous equation below and its corresponding homogeneous equation. (a) Find a general solution to the nonhomogeneous equation. (b) Find the solution that satisfies the specified initial conditions. x@y" + xy-y4 - Inxx>0; y(t) = 2, y(t) = 2.y"(1)=5: Yo-Inx-1: {x, xin x, xin x)} (a) Find a general solution to the nonhomogeneous equation yox) - CX+C_x Inx + CyX(In x)2 Inx-1 (b) Find the solution that satisfies the initial conditions y(1) -2.7(1)=2, and y" (1) - 5

Answer 1

Given differential equation x3 g" + y'"+ay'-y = 4- In a azo Particular solution Is Yp = lna - Fundamental solution Is y z cat candnat Cza (doa)? General solution of equation Genera)_golution - Fundamental solution + particular Solution : y(a) = catalna + c3a (Una) ²+ lox-1 b) Now we find solution with the initia) conditions YCb) = 2. Y'(1) = 2 y”(1) = 5 from o 9°W) = 6 1 636 2 + + Inse] + c3f22002740 Caf 14dna) Y'(x) = Cit C2/1t Inn + 203 Inatt x 2

22 Y"(x) = otca ( 1 ) + 2 C3 C2 1 + 263 | Y"(x)= (3 Given yo)=2 2 From o 90) = C (1) +G(1) do (1) + (3 (1) (do 1² 01- + Ins- C to foto-1 {. do (1) zo'? C 3 4'(1)=2 from ② y (1)= ci + C2(1 + In 1) +2 c3 {for 2+1 g 2=cit ce + 1 C+ (2 C2= 1-ci = 1-3=-2 [C2 = -2] Y" (1)=5 From ③ y") Cz + 2 (3-1 cz +2 C3 -1 (2 + 2c3 = 6 263 = 6-(-2) C3 = 4 Now solution is given by Cby o) 4(x) = 3a+ (-2) alnatya (anal2 + Ina-2