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Question
Solve` x^2 (d^3y)/dx^3+3x (d^2y)/dx^2+dy/dx+y/x=4log x`
Solution
`x^2 (d^3y)/dx^3+3x (d^2y)/dx^2+dy/dx+y/x=4log x`
The given diff. eqn is Cauchy’s homogeneous eqn .
Multiply the given eqn by x,
`x^3 (d^3y)/dx^3+3x^2 (d^2y)/dx^2+x dy/dx+y=4 x log x`
Put x = ๐๐ log x = z
Diff. w.r.t x,
`1/x=dz/dx` but `dy/dx=dy/dz.dz/dx=dy/dz 1/x`
∴ `x dy/dx=dy`
`x^2 (d^2 y)/dx^2=D(D-1)y`
`x^3 (d^3 y) /dx^3= D(D-1) (D-2)y`
∴ `[D(D-1)(D-2)+3D(D-1)+D+1]y=4z.e^z`
∴ `[D^3+1]y=4z.e^z`
For complementary solution ,
`f(D)=0`
∴` [D^3+1]y=4z.e^z`
Roots are: `D=-1,1/2+isqrt3/2,1/2-isqrt3/2`
Roots of the eqn are real and complex.
∴` y_c=c_1e^-z+e^(z/2) (c_2cos sqrt(3z)/2+c_3 sin sqrt(3z)/2)`
For particular integral ,
`y_p=1/f(D) x=1/(D^3+1) 4z.e^z`
= `4e^z 1/((D+1)^3+1) Z`
= `4e^Z 1/(D^3+3D^2+3D+2) z`
∴` y_p=e^z(2z-3)`
The general solution of given diff. eqn is ,
`y_g=y_c+y_p=c_1e_(-z/2)(c_2cos sqrt(3z)/2 + c_3 sin sqrt(3z)/2)+e^z(2z-3)`
Resubstitute z,
∴ `y_g=c_1/x + sqrtx(c_2cos sqrt(3 log x)/2+c_3 sin sqrt(3log x)/2)+(2 log x-3)`
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