W6 Second Order Differential Equations

Form

1. What is the form of 2nd ODE?
   • $F(x,y,\frac{dy}{dx},\frac{d^2y}{dx^2})$
2. What is the form of a Linear 2nd ODE?
   • $\frac{d^2y}{dx^2}+P(x)\frac{dy}{dx}+Q(x)y=R(x)$
3. When $R(x)=0$, the ODE is…
   • Homeogeneous, (H)
4. When $R(x)\ne0$, the ODE is…
   • Inhomeogeneous, (IH)

Homeogenous, Linear 2nd ODE

5. What is the general form of a homeogenous, linear 2nd ODE?
   • $ay''+by'+cy=0$
6. What is the general form of the general solution of a homeogenous, linear 2nd ODE?
   • $GS(H)=c_1y_1(x)+c_2y_2(x)$
7. How do you find the general solution of a homeogenous, linear 2nd ODE?
   • Step 1: Write the differential equation in the form $ay''+by'+cy=0$
   • Step 2: Find the corresponding characteristic equation $a\lambda^2+b\lambda+c=0$
     • Try $y(x)=e^{\lambda x}$
     • $\implies y'(x)=\lambda e^{\lambda x}$
     • $\implies y''(x)=\lambda^2 e^{\lambda x}$
     • $(a\lambda^2+b\lambda+c=0)e^{\lambda x}=0$
       • $e^{\lambda x}\ne0$
     • $a\lambda^2+b\lambda+c=0$ (Characteristic Equation)
     • $\lambda=\frac{-b\sqrt{b^2-4ac}}{2a}$ (Quadratic Equation)
   • Step 3: Determine the form of the general solution based on whether the characteristic equation has distinct, real roots; a single, repeated real root; or coplex conjugate roots.
   • Case 1: $b^2-4ac>0$
     • 2 distinct values: $\lambda_1, \lambda_2$
     • 2 linearly independent solution:
       • $y_1(x)=e^{\lambda_1x}$
       • $y_2(x)=e^{\lambda_2x}$
     • General Solution:
       • $y(x)=Ae^{\lambda_1x}+Be^{\lambda_2x}$, where $A, B\in\mathbb{R}$ constant and for all $x\in\mathbb{R}$
   • Case 2: $b^2-4ac=0$
     • 1 real value: $lambda=-\frac{b}{2a}$
     • solutions:
       • $e^{\lambda x}$
       • $xe^{\lambda x}$
     • General Solution
       • $y(x)=Ae^{\lambda x}+Bxe^{\lambda x}=(A+Bx)e^{\lambda x}$, where $A, B\in\mathbb{R}$ constants and for all $x\in\mathbb{R}$
   • Case 3: $b^2-4ac<0$
     • 2 complex conjugate values:
       • $\lambda_1=\alpha +i\beta$
       • $\lambda_2=\alpha -i\beta$
     • 2 complex linearly independent solutions
       • $y_1(x)=e^{(\alpha+i\beta)x}$
       • $y_2(x)=e^{(\alpha-i\beta)x}$
     • 2 real linearly independent solutions
       • $y_1(x)=e^{\alpha x}\cos(\beta x)$
       • $y_2(x)=e^{\alpha x}\sin(\beta x)$
     • General Solution:
       • $y(x)=Ae^{\alpha x}\cos(\beta x)+Be^{\alpha x}\sin(\beta x)$, where $A, B\in\mathbb{R}$ constants and for all $x\in\mathbb{R}$