Answer:
G.S=[tex]C_1\frac{1}{x}+C_2x^2+x^2logx+\frac{1}{2}[/tex]
Step-by-step explanation:
We are given that non-homogeneous differential equation
[tex]x^2y''-2y=3(x^2)-1[/tex]
It is Cauchy Euler equation
Substitute x=e^t x>0
Auxillary equation
[tex]D'(D'-1)-2=0[/tex]
[tex]D'^2-D'-2=0[/tex]
[tex](D'-2)(D'+1)=0[/tex]
[tex]D'-2=0 \implies D'=2[/tex]
[tex]D'+1=0\implies D'=-1[/tex]
Complementary solution
[tex]y=C_1e^{-t}+C_2e^{2t}[/tex]
[tex]y=C_1\frac{1}{x}+C_2x^2[/tex]
Particular solution
[tex]y_p=\frac{3e^{2t}}{D'^2-D'-2}-\frac{e^{0t}}{D'^2-D'-2}[/tex]
[tex]y_p=te^{2t}+\frac{1}{2}=x^2logx+\frac{1}{2}[/tex]
G.S=[tex]C_1\frac{1}{x}+C_2x^2+x^2logx+\frac{1}{2}[/tex]
Hence, general solution G.S=[tex]C_1\frac{1}{x}+C_2x^2+x^2logx+\frac{1}{2}[/tex]
