Two Linear ODEs — Variation of Parameters with a Secant Forcing, and a Resonant Polynomial+Exponential Forcing: 04-BS-1, May 2014, Q1

Question 1: Two Linear ODEs — Variation of Parameters with a Secant Forcing, and a Resonant Polynomial+Exponential Forcing (a) 10, (b) 10 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. (a) A linear, constant-coefficient ODE with a $\sec 3x$ forcing term (not a polynomial-times-exponential, so undetermined coefficients does not apply). (b) A linear, constant-coefficient ODE forced by a sum $3x^2+e^{-2x}$, where the exponential term's rate matches a homogeneous root.

Find. (a) The general solution $y(x)$. (b) The general solution $y(x)$.

Approach. (a) Solve the homogeneous equation, then use variation of parameters (the only method that handles $\sec$ forcing) with the Wronskian of $\cos 3x,\sin 3x$. (b) Split the forcing by superposition: a polynomial trial for $3x^2$, and — because $e^{-2x}$ duplicates the homogeneous root $r=-2$ — a resonance trial $Dxe^{-2x}$ for the exponential piece.

(a) Homogeneous solution and Wronskian. $r^2+9=0\Rightarrow r=\pm3i$, so $y_1=\cos3x,\ y_2=\sin3x$, and $$W=y_1y_2'-y_2y_1'=3\cos^23x+3\sin^23x=3.$$
(a) Variation-of-parameters integrals. With $f(x)=\sec3x$, $$\int\frac{y_2f}{W}dx=\int\frac{\sin3x\sec3x}{3}dx=\frac13\int\tan3x\,dx=-\frac19\ln|\cos3x|,$$ $$\int\frac{y_1f}{W}dx=\int\frac{\cos3x\sec3x}{3}dx=\int\frac13dx=\frac{x}{3}.$$
(a) Assemble $y_p$. $y_p=-y_1\!\int\!\tfrac{y_2f}{W}dx+y_2\!\int\!\tfrac{y_1f}{W}dx=\cos3x\cdot\tfrac19\ln|\cos3x|+\sin3x\cdot\tfrac{x}{3}$. $$y(x)=\boxed{C_1\cos3x+C_2\sin3x+\tfrac19\cos3x\ln|\cos3x|+\tfrac{x}{3}\sin3x}$$
(b) Homogeneous solution. $r^2-r-6=0=(r-3)(r+2)\Rightarrow r=3,-2$, so $y_h=C_1e^{3x}+C_2e^{-2x}$.
(b) Particular solution for $3x^2$. Try $y_{p1}=Ax^2+Bx+C$. Substituting and matching coefficients of $x^2,x^1,x^0$: $$-6A=3,\quad -2A-6B=0,\quad 2A-B-6C=0\ \Rightarrow\ A=-\tfrac12,\ B=\tfrac16,\ C=-\tfrac{7}{36}.$$ $$y_{p1}=-\tfrac12x^2+\tfrac16x-\tfrac{7}{36}.$$
(b) Particular solution for $e^{-2x}$ — resonance. Since $r=-2$ is already a homogeneous root, the plain trial $Ae^{-2x}$ fails; use $y_{p2}=Dxe^{-2x}$. Substituting, $$y_{p2}''-y_{p2}'-6y_{p2}=-5De^{-2x}\ \Rightarrow\ D=-\tfrac15.$$ $$y_{p2}=-\tfrac15xe^{-2x}.$$ $$y(x)=\boxed{C_1e^{3x}+C_2e^{-2x}-\tfrac12x^2+\tfrac16x-\tfrac{7}{36}-\tfrac15xe^{-2x}}$$

Part	Result
(a) $y(x)$	$C_1\cos3x+C_2\sin3x+\tfrac19\cos3x\ln\|\cos3x\|+\tfrac{x}{3}\sin3x$
(b) $y(x)$	$C_1e^{3x}+C_2e^{-2x}-\tfrac12x^2+\tfrac16x-\tfrac{7}{36}-\tfrac15xe^{-2x}$

Concept & Theory

Topic. Variation of parameters (for non-polynomial-exponential forcing) and resonant undetermined coefficients.

Key relations. For $y''+p y'+qy=f$ with known homogeneous solutions $y_1,y_2$ and Wronskian $W$, $y_p=-y_1\int\frac{y_2f}{W}dx+y_2\int\frac{y_1f}{W}dx$. If a forcing term $e^{kx}$ (or $x^ne^{kx}$) duplicates a homogeneous solution, multiply the usual trial by $x$ (or by $x^m$ for an $m$-fold root) until independence is restored.

Why this works. $\sec3x$ has no finite family of derivatives closed under differentiation, so undetermined coefficients cannot supply a trial — variation of parameters instead builds $y_p$ directly from the known homogeneous basis and works for any continuous forcing. In (b), superposition lets the polynomial and exponential pieces of the forcing be solved independently and added.

Common pitfall. Dropping the absolute value inside $\ln|\cos3x|$ (needed for real logarithms when $\cos3x<0$), and, in (b), using the non-resonant trial $Ae^{-2x}$ which collapses to $0=e^{-2x}$ on substitution.

Source. Kreyszig, Advanced Engineering Mathematics, Ch. 2 (§2.10 Variation of Parameters; §2.7 Undetermined Coefficients — resonance).

Question 1 of 8: Two Linear ODEs — Variation of Parameters with a Secant Forcing, and a Resonant Polynomial+Exponential Forcing

Question 1: Two Linear ODEs — Variation of Parameters with a Secant Forcing, and a Resonant Polynomial+Exponential Forcing (a) 10, (b) 10 marks