How to solve x^2 y'' - (x^2 +2x)y' + (x+2)y = x^3 e^x

What do we do when there is no hint given?

how to guess the probable solution, can we use the method used in the method of undetermined coefficients?

Printable View

- October 24th 2012, 11:10 AMsshsolving Second order non - homogeneous Differential Equation
How to solve x^2 y'' - (x^2 +2x)y' + (x+2)y = x^3 e^x

What do we do when there is no hint given?

how to guess the probable solution, can we use the method used in the method of undetermined coefficients? - October 24th 2012, 06:13 PMProve ItRe: solving Second order non - homogeneous Differential Equation
Because it does not have constant coefficients, I expect you need to look for a series solution...

- October 25th 2012, 04:24 AMsshRe: solving Second order non - homogeneous Differential Equation
But this question appears in the book before the power series and other questions have a hint given. then assuming the hint as y1, y2 = vy1. But this question no hint is given, in this case how can we guess a solution.

- October 25th 2012, 05:10 AMJJacquelinRe: solving Second order non - homogeneous Differential Equation
Since the term on right side is on the form P(x)*exp(x) with P(x) a polynomial (in the present case P(x)=x^3), then we guess that y(x) could be on the form :

y(x) = p(x)*exp(x) where p(x) is a polynomial.

Considering the term on the left of the ODE, it appears that the degree of P(x) is the degree of p(x)+one unit.

So, we guess that y(x)=p(x)*exp(x) where p(x) is a polynomial of degree 2.

Then, the coefficients of this polynomial are obtained by identification.

On the other hand, it is not difficult to find that y=x and y=x*exp(x) are independant solutions of the ODE without the term on the right side.