Nonhomogeneous Euler–Cauchy Equation: 04-BS-1, May 2018, Q1

Question 1: Nonhomogeneous Euler–Cauchy Equation (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. An Euler–Cauchy (equidimensional) ODE $x^2y''-4xy'+6y=3x^4$.

Find. The general solution $y(x)$ (two arbitrary constants).

Approach. Solve the homogeneous equation with the power trial $y=x^m$, then, since the forcing power $x^4$ is not one of the homogeneous roots, find a particular solution with the matching trial $y_p=Ax^4$.

Homogeneous solution. Substituting $y=x^m$: $x^2m(m-1)x^{m-2}-4xmx^{m-1}+6x^m=[m(m-1)-4m+6]x^m=0\Rightarrow m^2-5m+6=0=(m-2)(m-3)$, so $m=2,3$ and $$y_h(x)=C_1x^2+C_2x^3.$$
Particular solution. Since $x^4$ is not an Euler-compatible root here, try $y_p=Ax^4$: $y_p'=4Ax^3,\ y_p''=12Ax^2$. Substituting, $$x^2(12Ax^2)-4x(4Ax^3)+6(Ax^4)=(12A-16A+6A)x^4=2Ax^4=3x^4\ \Rightarrow\ A=\frac32.$$
Assemble the general solution. $$y(x)=C_1x^2+C_2x^3+\frac32x^4.$$

$$y(x)=\boxed{C_1x^2+C_2x^3+\tfrac32x^4}$$

Quantity	Result
Homogeneous roots	$m=2,3$
Particular solution	$\tfrac32x^4$
$y(x)$	$C_1x^2+C_2x^3+\tfrac32x^4$

Concept & Theory

Topic. Undetermined coefficients for a nonhomogeneous Euler–Cauchy equation.

Key relations. For $x^2y''+pxy'+qy=0$ the power trial $y=x^m$ gives the auxiliary equation $m(m-1)+pm+q=0$; when the forcing is $cx^n$ with $n$ not a root, the particular trial $y_p=Ax^n$ works directly (no resonance factor of $\ln x$ needed).

Why this works. Every term of an Euler–Cauchy operator applied to $x^n$ returns a multiple of $x^n$, so plugging $y_p=Ax^n$ collapses the ODE to a single algebraic equation for $A$ whenever $n$ avoids the two homogeneous roots $2,3$ — here $n=4$ is clear of both.

Common pitfall. Using the constant-coefficient trial $Ae^{4x}$ instead of the Euler-compatible $Ax^4$; forgetting to check whether $n$ coincides with a root (which would require $y_p=Ax^n\ln x$ instead).

Source. Kreyszig, Advanced Engineering Mathematics, Ch. 2 (§2.5 Euler–Cauchy Equations).

Question 1 of 8: Nonhomogeneous Euler–Cauchy Equation

Question 1: Nonhomogeneous Euler–Cauchy Equation (20 marks)