Question

Find a particular solution, given the fundamental set of solutions of the complementary equation. $$ x^{3} y^{\prime \prime \prime}-3 x^{2} y^{\prime \prime}+6 x y^{\prime}-6 y=2 x ; \quad\left\\{x, x^{2}, x^{3}\right\\} $$

Short answer

Question: Find the particular solution to the given differential equation using the method of undetermined coefficients: $$x^{3} y^{\prime \prime \prime}-3 x^{2} y^{\prime \prime}+6 x y^{\prime}-6 y=2x$$ Answer: The particular solution to the given differential equation is $$Y_p = -\dfrac{1}{3}x$$.

Step-by-step solution

Determine the General Form of the Particular Solution

For the given differential equation, $$ x^{3} y^{\prime \prime \prime}-3 x^{2} y^{\prime \prime}+6 x y^{\prime}-6 y=2 x $$ the right-hand side is a polynomial of degree 1 (2x). Therefore, we assume that the particular solution is also a polynomial of degree 1. Let the general form of the particular solution be: $$ Y_p = Ax + B $$ where A and B are constants to be determined.

Calculate the Derivatives

Next, to find the value of the undetermined coefficients A and B, calculate the first, second, and third derivatives of Y_p with respect to x: $$ Y_p^\prime = A $$ $$ Y_p^{\prime\prime} = 0 $$ $$ Y_p^{\prime\prime\prime} = 0 $$

Substitute the Particular Solution and its Derivatives into the Equation

Now, substitute Y_p and its derivatives into the given differential equation: $$ x^3(0) - 3x^2(0) + 6x(A) - 6(Ax + B) = 2x $$ which simplifies to: $$ 6Ax - 6Ax - 6B = 2x $$

Solve for the Coefficients A and B

Since the equation simplifies to: $$ -6B = 2x $$ we can directly solve for B: $$ B = -\dfrac{1}{3}x $$ As the equation simplified to such a state that the x-terms vanished, it means that the particular solution has a pure constant form, that is, A=0. So the particular solution to the given complementary equation is: $$ Y_p = -\dfrac{1}{3}x $$

Key concepts

Complementary Equation

Understanding the existence of a complementary equation is crucial when solving differential equations. It represents the homogenous part of the equation, without the nonhomogenous term (in this case, without the term '2x'). The solutions to the complementary equation are fundamental to the overall solution and are called the fundamental set of solutions. In the given problem, the fundamental set of solutions is \( \{x, x^2, x^3\} \), indicating that these functions solve the homogenous part of the differential equation.

The complementary equation plays a vital role in finding the general solution to the nonhomogenous differential equation. In this context, it serves as the foundation from which we can seek additional specific solutions driven by the nonhomogenous component ('2x'). The complementary solution alone would not suffice, which is why we turn to the method of undetermined coefficients to find the particular solution that complements it.

Undetermined Coefficients

The method of undetermined coefficients is a systematic way to find specific solutions to nonhomogenous differential equations. In our exercise, this approach helps us determine the particular coefficients (A and B) for the nonhomogenous term '2x'.

We expect the particular solution to have a similar form to the nonhomogeneous term. Since '2x' is a polynomial of the first degree, we hypothesize that the particular solution \( Y_p \) is also a polynomial of the first degree, \( Y_p = Ax + B \). After this assumption, the next step involves calculating the derivatives of \( Y_p \) and plugging them into the differential equation. This substitution will yield an algebraic equation with the undetermined coefficients A and B, which we can solve to find the specific values that make up our particular solution.

Polynomial Particular Solution

The selection of a polynomial form for the particular solution is based on the behavior of the nonhomogeneous term in the differential equation. If the nonhomogeneous term is a polynomial, as in the term '2x' from the given problem, we try a polynomial of the same degree for our particular solution. This process is applied because differential operators applied to polynomials yield polynomials of the same or lower degree.

In our case, by calculating the derivatives of \( Y_p \) and substituting them into the original equation, we discover that A must be zero to satisfy the equation. Essentially, the presence of the terms \( x, x^2, \) and \( x^3 \) in the fundamental set suggests that our initial assumption should have been adjusted to prevent duplication with the complementary solution; this mechanical error highlights the importance of examining the entirety of the complementary equation before proposing a form for the particular solution.

Eventually, by solving the resulting algebraic equation, we conclude that the particular polynomial solution for the given differential equation is \( Y_p = -\frac{1}{3}x \), demonstrating that the nonhomogeneous term '2x' only influences the constant term B in the particular solution.