How to Find Constants for the IVP Solution?

[find_the_fun]

The given family of functions is the general solution of the D.E. on the indicated interval. Find a member of the family that is a solution of the initial-value problem.

[math]y=c_1x+c_2x\ln{x}[/math] on [math](0, \infty)[/math] and [math]x^2y''-xy'+y=0[/math] and y(1)=3, y'(1)=-1

So plugging in y(1)=3 gives [math]3=c_1+c_2\ln{1}[/math] and then take the derivative to get [math]y'=c_1+c_2 \ln{x} +c_2[/math] subbing in [math]-1=C-1+c_2\ln{1}+c_2[/math]

adding 3 times the second equation to the first give [math]0=4c_1+4c_2\ln{1}+3c_2[/math]
What next?

 

[MarkFL]

Okay, we have:

$$y(1)=c_1=3$$ (recall $\log_a(1)=0$)

$$y'(1)=c_1+c_2=-1$$

Now the system is easier to solve. :D

 

[find_the_fun]

Okay, we have:

$$y(1)=c_1=3$$ (recall $\log_a(1)=0$)

$$y'(1)=c_1+c_2=-1$$

Now the system is easier to solve. :D

Ok but we never used [math]x^2y''-xy'+y=0[/math] Why was that in the question?

 

[MarkFL]

Ok but we never used [math]x^2y''-xy'+y=0[/math] Why was that in the question?

That was to show you which ODE the given solution satisfies. You are right though, we didn't need it to find the particular solution satisfying the given conditions. The given ODE is a Cauchy-Euler equation which may be solved either by making the substitution:

$$x=e^t$$

which will given you a homogeneous equation with constant coefficients, or by assuming a solution of the form:

$$y=x^r$$

which will give you an indicial equation with a repeated root to which you may apply the method of reduction of order.