Question

Let L(y)=an​y(n)(x)+an−1​y(n−1)(x)+⋯+a1​y′(x)+a0​y(x) where a0​,a1​,…,an​ are fixed constants. Consider the nth order linear differential equation L(y)=3e6xcosx+5xe6x Suppose that it is known that L[y1​(x)]=6xe6x when y1​(x)=24xe6x L[y2​(x)]=5e6xsinx when y2​(x)=40e6xcosx L[y3​(x)]=4e6xcosx when y3​(x)=8e6xcosx+16e6xsinx Find a particular solution to (∗).

Answer 1

The student seeks a particular solution to a linear differential equation using given solutions under the L operator, applying superposition combined with known resultant expressions.

Step-by-step explanation:

The student is dealing with a linear differential equation of the form L(y) which includes various constants and functions of y. Given particular solutions for variants of L(y), the objective is to find a particular solution to the modified differential equation L(y) = 3e6xcosx + 5xe6x. Since the operator L seems to act linearly on the provided solutions, we can use the principle of superposition to construct the particular solution. This is done by combining the existing solutions y1(x), y2(x), and y3(x), with their respective outputs after the L operator has been applied, to match the right-hand side of the given differential equation.