Thread: Differential Equation Question

Hi. I have the equations y= Cx^2 - 2x and y' - 2(y/x)=2, and y(1)=10. The question reads "Verify that y is a general solution of the differential equation. Then find a particular solution of the differential equation that satisfies the given point (1,10)".

I was able to find the particular solution (y=12x^2 - 2x), but was not able to "Verify that y is a general solution of the differential equation".

What I did was have y'=2Cx - 2, and then substituted this into y' - 2 (y/x)=2 to get (2CX-2) - 2y/x =2, making (2Cx^2 - 2y)/(x^2)=4. Where do I go from here?

Thanks
Jay

Originally Posted by jaijay32

Hi. I have the equations y= Cx^2 - 2x and y' - 2(y/x)=2, and y(1)=10. The question reads "Verify that y is a general solution of the differential equation. Then find a particular solution of the differential equation that satisfies the given point (1,10)".

I was able to find the particular solution (y=12x^2 - 2x), but was not able to "Verify that y is a general solution of the differential equation".

What I did was have y'=2Cx - 2, and then substituted this into y' - 2 (y/x)=2 to get (2CX-2) - 2y/x =2, making (2Cx^2 - 2y)/(x^2)=4. Where do I go from here?

Thanks
Jay

In addition to substituting 2Cx- 2 for y', substitute Cx^2- 2x for y.
y'- 2y/x= (2Cx- 2)- (2Cx^2- 4x)/x= 2Cx- 2- 2Cx+ 4= 2.

(You have an error in what you did- (2Cx- 2)- 2y/x= (2Cx^2- 2x- 2y)/x. You dropped the "-2x".)