Method of Inspection

[Hong]

Hi,

I was wondering if someone can me help me solve this 2nd order linear, homogeneous differential equation:

r^2*(f'')+(2r)*(f')-2f=0

The general solution has to be in the form of: f(r)=c1*f1(r)+c2*f2(r)

By method of inspection, a solution is: f1(r)=r

So therefore f2=g(r)*f1]

where g(r) is some other function of r.

My question is, how do you use method of inspection to figure out a solution? Can anyone direct me to some other resouces that explain examples like this: 2nd order equations with a variable in front of the second derivative ex. (like x^2 in front of the 2nd derivative)

Thanks.

 

[DrMike]

"Method of inspection" is another name for "guess a solution".

More sophistactedly, you might "guess" a parametrised solution, and solve for the parameter. Eg, y''+8y'+15y=0, we 'guess' y=exp(kx), and solve for k.

Or, for x^2y''+4xy'+7x^2=x^3, we might 'guess' y=Ax^b...
plugging this in gives Ab(b-1)x^b + 4Abx^b+7Ax^b=x^3, so, A(b(b-1)+4b+7)x^b=x^3.
Then solve for A and b... it seems clear that b=3, then we get A(12-12+7)=1, so A=1/7.