D Operators - Differential Equations

Question

Using D Operator method find the general solution of the differential equation,

d^y/dx^2 - 2dy/dx + 2y = 1

Answer 1

Rewrite dy²/dx²−2dy/dx+2y=1 as D²y−2Dy+2y=1 where D²y & Dy denote dy²/dx² & dy/dx.

For D²y−2Dy+2y=1 expressed as (D²−2D+2)y=1, solve the Characteristic Equation D²−2D+2=0

and obtain D=[-(-2)±√((-2)²-4(1)(2))]/2(1) which amounts to 1±i. For Complex Conjugates 1+i and 1-i,

it can be shown for the Linear Homogeneous Equation D²y−2Dy+2y=0 that the Homogeneous Solution

can follow the form y_h= e^(1)x(c₁cos (1)x+c₂sin (1)x); the ones in parentheses come from the Complex

Conjugates (1)±(1)i above.

Next assume a Particular Solution y_p=A₀. Since y_h has no terms in common with y_p, no modification of

y_p is necessary. For the Constant A_0, it follows that y'_p and y''_p are both 0.

Use of y_p in (D²−2D+2)y=1 will then give (0²−2(0)+2)A₀=1 or A₀=1/2.

Finally, write the General Solution as y=y_h + y_p or e^x(c₁cos x+c₂sin x) + 1/2.