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yKpoI14uk [10]
3 years ago
11

Find the general solution of the nonhomogeneous differential equation x^2y''-2y=3(x^2) -1, (x>0).

Mathematics
1 answer:
Ira Lisetskai [31]3 years ago
6 0

Answer:

G.S=C_1\frac{1}{x}+C_2x^2+x^2logx+\frac{1}{2}

Step-by-step explanation:

We are given that non-homogeneous differential equation

x^2y''-2y=3(x^2)-1

It is Cauchy Euler equation

Substitute x=e^t  x>0

Auxillary equation

D'(D'-1)-2=0

D'^2-D'-2=0

(D'-2)(D'+1)=0

D'-2=0 \implies D'=2

D'+1=0\implies D'=-1

Complementary solution

y=C_1e^{-t}+C_2e^{2t}

y=C_1\frac{1}{x}+C_2x^2

Particular solution

y_p=\frac{3e^{2t}}{D'^2-D'-2}-\frac{e^{0t}}{D'^2-D'-2}

y_p=te^{2t}+\frac{1}{2}=x^2logx+\frac{1}{2}

G.S=C_1\frac{1}{x}+C_2x^2+x^2logx+\frac{1}{2}

Hence, general solution G.S=C_1\frac{1}{x}+C_2x^2+x^2logx+\frac{1}{2}

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