Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation \( y^{\prime \prime}-16 y=x \sin x \) \( y_{p}=(A x+B) \cos x+(C x+E) \sin x \) \( y_{p}=(A x+B) \cos x \) \( A x \cos x+B \sin x \) \( y_{p}=(A x+B) \sin x \) \( y_{p}=A \cos x+B \sin x \)

To find the correct form of a particular solution for the differential equation \( y^{\prime \prime}-16 y=x \sin x \), we first look at the non-homogeneous part, which is \( x \sin x \). Given this part involves \( x \) multiplied by \( \sin x \), we will consider that in our form for the particular solution. The correct approach includes terms that account for both components, hence, \( y_{p}=(A x+B) \cos x+(C x+E) \sin x \) captures both the sine and cosine components along with the polynomial part of degree one. Now, let’s explore a related concept! The method of undetermined coefficients is all about matching your guess to the right side of the differential equation. Having \( x \sin x \) suggests polynomial multiplication will be necessary, which is why we include those \( Ax + B \) types of terms to counteract that polynomial growth. Ready for a deeper dive? You might find it fascinating how variations of this method can handle even more complex non-homogeneous equations. If you're intrigued, look into the method of variation of parameters! It’s a brilliant way to derive solutions without guessing a particular form.