solve by using variation of parametersby SOS2012 Tags: complementary, differential, general solution, variation parameters 

#1
Nov1912, 03:21 PM

P: 4

x²y"(x)3xy'(x)+3y(x)=2(x^4)(e^x)
=>y"(x)(3/x)y'(x)+(3/x²)y(x)=2x²e^x i dont know how to approach this problem because the coefficients are not constant and i am used to being given y1 and y2 HELP!!! 



#2
Nov1912, 04:33 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,898

This is an "Euler type" or "equipotential" equation. The substitution t= ln(x) will change it to a "constant coefficients" problem in the variable t.
[tex]\frac{dy}{dx}= \frac{dy}{dt}\frac{dt}{dx}= \frac{1}{x}\frac{dy}{dt}[/tex] and, differentiating again, [tex]\frac{d^2y}{dx^2}= \frac{d}{dx}\left(\frac{dy}{dx}\right)[/tex][tex]= \frac{d}{dx}\left(\frac{1}{x}\frac{dy}{dt}\right)= \frac{1}{x^2}\frac{dy}{dt}+ \frac{1}{x^2}\frac{d^2y}{dt^2}[/tex] 



#3
Nov1912, 04:41 PM

P: 4




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