# Solve X 2 D 3 Y D X 3 + 3 X D 2 Y D X 2 + D Y D X + Y X = 4 Log X - Applied Mathematics 2

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 = 𝒆𝒛 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)

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
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