Answer to Question #196647 in Differential Equations for Atharv Upadhyay

Question

(D²-2DD'+D'²)z=cosy-xsiny

Answer 1

First, we consider that D = m and D' = 1 to find the characteristic equation and then we find out the type of equation or case that we need to solve:

(D²-2DD'+D'²)z=(m² - 2m + 1)z = cos y - x*sin y

Then, from m² - 2m + 1 = 0 = (m - 1)², this means that m₁ = m₂ = 1 the complementary solution has the form

Z_{complementary} = f₁(y+x) + x*f₂(y+x)

The general solution is found by analyzing the particular integral PI and using c = y + x to find the complete solution on this integral (that has an auxiliary equation with repeated root) we find:

$PI=\frac{1}{D^2-2DD' + D´^2}\int (cos\,y\,-xsin\,y)$

$P.I.= \frac{1}{(D-D´)^2} \int (\int (cos\,(c-x)\,-xsin\,(c-x)) dx)dx$

$PI = xsin(c-x)-3cos(c-x)=xsiny-3cosy$

Thus, the complete solution for Z is the sum of the complementary and the particular solution:

$Z = f_1(y+x)+xf_2(y+x)+xsiny-3cosy$