3 L13: Non Homogeneous Linear Equations (5.3)
3.1 Non Homogeneous Linear Equations
\[y'' + p(x)y'+q(x)y=f(x)\]
\(f(x)\neq 0\)
Examples:
- Forced Harmonic Oscillator with damping: \[\boxed{m x'' + b x' + k x = F(t)}\]
- Electrical circuits: \[\boxed{Lq''+Rq'+(1/C) q = V(t) }\]
3.2 Theorem Existence & Uniqueness
Theorem 3.1 (Existence & Uniqueness) Suppose \(p,q,f\) are continuous on \((a,b)\). Let \(x_0\in(a,b),\,k_0,k_1\in\mathbb{R}\)
Then the initial value problem \[y''+p(x)y'+q(x)y=f(x),\qquad y(x_0)=k_ 0,\quad y'(x_0)=k_1\] has a unique solution on \((a,b)\)
3.3 The Complementary Equation
\[y''+p(x)y'+q(x)y=f(x)\] has a complementary equation
\[y''+p(x)y'+q(x)y=0\]
3.4 General Sol for Non-Homogeneous
Theorem 3.2 (General Solution and complementary equation) Suppose \(p,q,f\) are continuous on \((a,b)\).
Let \(y_p\) be a particular solution to
\[y'' + p(x)y'+q(x)y=f(x) \tag{3.1}\]
Let \(\{y_1,y_2\}\) be a fundamental set of solutions to the complementary equation \[y'' + p(x)y'+q(x)y=0 \tag{3.2}\] Then the general solution to Equation 3.1 is
\[y=c_1y_1+c_2y_2+y_p\]
Proof.
\[y'=c_1y_1'+c_2y_2'+y_p'\] \[y''=c_1y_1''+c_2y_2''+y_p''\]
\(\Rightarrow\) plug into the 2OLNH (Equation 5.1):
\[\begin{align} (c_1y_1''+c_2y_2''+y_p'') +&\\ p(x)(c_1y_1'+c_2y_2'+y_p')+\\ q(x)(c_1y_1+c_2y_2+y_p)&\\ &=f(x) \end{align}\]
\[\begin{align} c_1(y_1''+p(x)y_1'+q(x)y_1)&\\ c_2(y_2''+p(x)y_2'+q(x)y_2)+\\ y_p''+p(x)y_p'+q(x)y_p&\\ &=f(x) \end{align}\] \[\square\]
Solve \[y''+y=1,\qquad y(0)=2,\quad y'(0)=7\]
Complementary Equation: \[\Rightarrow y''+y=0\] \[r^2+1=0\Rightarrow r_{1,2}=0\pm 1i\] \[y_i=c_1\cos x +c_2\sin x\]
Particular Solution: (By inspection\(\rightarrow\) techniques to follow) \[y=1\]
General Solution: \[y=c_1\cos x +c_2\sin x+1\]
Initial Conditions: \[y(0)=2\Rightarrow c_1+1=2\Rightarrow c_1=1\] \[y'=-\sin x +c_2\cos x\Rightarrow y'(0)=7\Rightarrow c_2=7\]
Final Solution:
\[\boxed{y=\cos x+ 7\sin x +1}\]
- Find the general solution of \[y'' − 2y' + y = −3 − x + x^2\]
- Solve the initial value problem \[y'' − 2y' + y = −3 − x + x^2,\qquad y(0) = −2,\quad y'(0) = 1\]
Let’s find first the solution for the homogeneous, where the characteristic equation is:
\[r^2-2r+1=0\] Then \[r_{1,2}=\frac{+2\pm\sqrt{4-4}}{2}=1=m\rightarrow \boxed{m=-p/2=1}\]
Therefore:
\[y=c_1e^{x}+c_2xe^{x}+y_p\] Now the goal is a particular for the non homogeneous:
\[y'' − 2y' + y = −3 − x + x^2\] Guess:
Let’s choose \(y_p\) as a polynomial of order 2, so the LHS have the same order of RHS:
\[\boxed{y_p=Ax^2+Bx+C}\rightarrow y'_p=2Ax+B\rightarrow y''_p=2A\] and see what are the conditions the constants need to satisfy:
\[2A-2(2Ax+B)+(Ax^2+Bx+C)=−3 − x + x^2\] Grouping:
\[(2A-2B+C) + (-4A+B)x +Ax^2=−3 − x + x^2\] Therefore:
\[\begin{align} 2A-2B+C&=-3\\ -4A+B&=-1\\ A&=1\\ \end{align}\]
\[\Rightarrow \boxed{A=1}\rightarrow \boxed{B=3}\rightarrow \boxed{C=1}\]
Then a particular soln for the non-homogeneous:
\[\boxed{y_p= x^2+3x+1}\]
Then the general solution for the non-homogeneous is:
\[\boxed{\boxed{y=c_1e^{x}+c_2xe^{x}+x^2+3x+1}}\]
\[y(0) = −2,\quad y'(0) = 1\] \[y=c_1e^{x}+c_2xe^{x}+x^2+3x+1\] i.c.1: \[-2=c_1+0+0+0+1\Rightarrow \boxed{c_1=-3}\] i.c.2:
\[y=-3e^{x}+c_2xe^{x}+x^2+3x+1\] \[\Rightarrow\] \[y'=-3e^{x}+c_2e^{x}+c_2xe^{x}+2x+3\]
\[1=-3+c_2+c_2\cdot 0+2\cdot 0+3\] Then \[1=-3+c_2+3\rightarrow \boxed{c_2=1}\]
Finally, the unique solution for this initial conditions is:
\[\boxed{y=-3e^{x}+xe^{x}+x^2+3x+1}\]
3.5 Particular solution as a combination of particulars
Theorem 3.3 (Particular Solutions) If \(y_{p_1}\) is a particular solution of
\[y'' +p(x)y' + q(x)y =f_1(x)\] and \(y_{p_2}\) is a particular solution of
\[y'' +p(x)y' + q(x)y =f_2(x)\] Then \(\boxed{y_p=y_{p_1}+y_{p_2}}\) is a particular solution of
\[y''+p(x)y'+q(x)y=f_1(x)+f_2(x)\]
Proof. \[y''+p(x)y'+q(x)y\stackrel{?}{=}\] \[(y_{p_1}''+y_{p_2}'')+p(x)(y_{p_1}'+y_{p_2}')+q(x)(y_{p_1}+y_{p_2})\stackrel{?}{=}\] \[(y_{p_1}''+p(x)y_{p_1}'+q(x)y_{p_1})+(y_{p_2}''+p(x)y_{p_2}'+q(x)y_{p_2})\stackrel{?}{=}\] \[=f_1(x)+f_2(x)\] \[\square\]
3.7 References
- https://quarto.org/docs/authoring/cross-references.html#theorems-and-proofs
- https://quarto.org/docs/presentations/revealjs/advanced.html
- https://quarto.org/docs/presentations/revealjs/presenting.html#chalkboard
- LibreTexts, 5.3: Nonhomgeneous Linear Equations from William F. Trench.
- LibreTexts, 17.2: Nonhomogeneous Linear Equations from Gilbert Strang & Edwin “Jed” Herman OpenStax.
- Chatgpt: Physics Examples of NonHomogeneous Linear Equations
- LibreText 23.6: Forced Damped Oscillator from Peter Dourmashkin MIT
- Undetermined coefficients 1 | Second order differential equations | Khan Academy
later we will show how and why.↩︎