Question #299374

Solve:


x².d²y/dx² -x.dy/dx +y=ln x

1
Expert's answer
2022-02-22T10:47:11-0500

Let us use the transformation x=et.x=e^t. Then 

yx=ytet, yx2=(yt2yt)e2t.y'_x=y'_te^{-t},\ y''_{x^2}=(y''_{t^2}-y'_t)e^{-2t}.


We get the equation for the function y(t)y(t)

e2t(yy)e2tetyet+y=t,e^{2t}(y''-y')e^{-2t} -e^ty'e^{-t} +y=t,

 which is equivalent to 

y2y+y=ty''-2y'+y=t


Corresponding homogeneous differential equation


y2y+y=0y''-2y'+y=0

Characteristic (auxiliary) equation


r22r+1=0r^2-2r+1=0

r1=r2=1r_1=r_2=1

The general solution of the homogeneous differential equation is


yh=c1et+c2tety_h=c_1e^t+c_2 te^t


Find the particular solution of the non homogeneous differential equation


yp=At+By_p=At+B

yp=Ay_p '=A

yp=0y''_p=0

Substitute


02A+At+B=t0-2A+At+B=t

A=1,B=2A=1, B=2

The general solution of the non homogeneous differential equation is


y=c1et+c2tet+t+2y=c_1e^t+c_2 te^t+t+2

y(x)=c1x+c2xlnx+lnx+2y(x)=c_1x+c_2x\ln x+\ln x+2


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS