Differential Equations Exam 2

What is the particular solution to the equation(Use Letters): y”+ay=x²

Ax² + Bx + C

What is the particular solution to the equation(Use Letters): y”+ay=-cos(3t)

A cos(3t) + B sin(3t)

What is the particular solution to the equation(Use Letters): y”+ay=e^(2t)

A e^(2t)

What is the particular solution to the equation(Use Letters): y”+ay=te^(2t)

A te^(2t) + B e^(2t)

What is the particular solution to the equation(Use Letters): y”+ay=t²e^(2t)

A t²e^(2t) + B te^(2t) + C e^(2t)

What is the particular solution to the equation(Use Letters): y”+ay=sin(3t)

A sin(3t) + B cos(3t)

What is the particular solution to the equation(Use Letters): y”+ay=4t³sin(3t)

t(At³+Bt²+Ct+D)sin(3t) + t(Et³+Ft²+Gt+H)cos(3t)

The general equation is:

y_c+y_p

What is case 1 for homogeneous equations and what does y_{c equal}

Case 1 pertains to situations where the characteristic equation yields two distinct real roots.

C₁e^r₁x+C₂e^r₂x

What is case 2 for homogenous equations and what is yc

Case 2 arises when the characteristic equation has repeated real roots.

(C₁+C₂t)e^rt

What is case 3 for homogenous equations and what is yc

Case 3 occurs when the characteristic equation has complex conjugate roots.

e^at(C₁cos(Bx) + C₂sin(Bx))