Question

Find the general solution, given that y1=ex satisfies the complementary equation. Use c1 and c2 for the two constants. xy′′−(2x+1)y′+(x+1)y=−ex y= Find the general solution, given that y1=x2 satisfies the complementary equation. Use c1 and c2 for the two constants. x2⋅y′′−5x⋅y′+8y=4x2 y=

Answer 1

The general solutions for the given differential equations are expressed as y=c1*y1 + yp, where y1 is a solution to the complementary equation, yp is a particular solution, and c1 is a constant.

Step-by-step explanation:

Given that y1=ex satisfies the complementary equation of the differential equation xy''-(2x+1)y\'+(x+1)y=-ex, the general solution to the nonhomogeneous differential equation can be expressed as y=c1y1 + yp, where yp is a particular solution to the nonhomogeneous equation and c1 is a constant.

For the second differential equation x2y''-5xy\'+8y=4x2, since y1=x2 satisfies the complementary equation, the general solution can be written as y=c1y1 + yp with c1 as a constant and yp as the particular solution related to the term 4x2.