Solve Differential Equation: x^2d^2y/dx^2+3xdy/dx+5y=3x

[fan_103]

$$x^2$$$$d^2$$y/$$dx^2$$+$$3x$$$$dy$$/$$dx$$+$$5y$$=$$3x$$I don't know where to start with the question ,can anyone here help me pleasez.

 

[pbandjay]

This equation looks to be solvable using the variation of parameters method or a green's function. To rearrange:

$$y''+3x^{-1}y'+5x^{-2}y=3x^{-1}$$

The method states that if you have

$$y''+a_1(x)y'+a_2(x)y=F$$

then let y₁ and y₂ be solutions to the associated homogeneous equation.

Then the particular solution is

$$y_p=u_1y_1+u_2y_2$$

where u₁ and u₂ satisfy both:

$$y_1u_1'+y_2u_2'=0$$

$$y_1'u_1'+y_2'u_2'=F$$

and the general solution:

$$y(x)=c_1y_1+c_2y_2+y_p$$

To use a green's function to find y_p, then

$$y_p(x)=\displaystyle\int_{x_0}^xK(x,t)F(t)dt$$

where the green's function, K(x,t), is defined as

$$K(x,t)=\frac{y_1(t)y_2(x)-y_2(t)y_1(x)}{W[y_1,y_2](t)}$$

and the Wronskian, W[y₁,y₂](t), is defined as

$$W[y_1,y_2](t)=\begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix}=y_1y_2'-y_2y_1'$$

unless I made a typo somewhere.

 

[fan_103]

Wow U r legend!Thanks a lot matey ;)