Question

Answer Problems 1-10 without referring back to the text. Fill in the blank or answer true or false. For the method of undetermined coefficients, the assumed form of the particular solution y_p for y^¢¢-y = 1+e⁸is ____

Short answer

$y_p=-A+Bx{e}^x$

Step-by-step solution

Definition

A Particular Solution of a differential equation is a solution obtained from the General Solution by assigning specific values to the arbitrary constants.

For particular solution

From the superposition principle for non-homogeneous equations, divide y_p;

$y_p=y_{p1}+y_{p2}{y}_{p1}$for the constant part and $y_{p2}$is for exponential part of the equation.

General assumption of y_p1 is A, general assumption of y_p2is ${\mathrm{Be}}^{\mathrm{x}}$

Substitute and get contradictory statement - A = 1 + e^x

This implies the assumption of y_p2 is wrong.

Observe that our assumption Be^x is already present in y_c. This means that e^x is a solution of the associated homogeneous differential equation, and a constant multiple Be^x when substituted into the differential equation necessarily produces zero.

That implies,$y_p=-A+Bx{e}^x$