Second Order Linear DE

$$ay''+by'+cy=0$$  

$$ay''+by'+cy=Q$$  

$$y''+by'+cy=0$$  

$$y''+by'+cy=Q$$  

$$a=0$$  

$$b=0$$  

$$c=0$$  

$$Q=0$$  

$$y=y_c+y_p$$  

$$y=C_1e^{\alpha x}+C_2e^{\beta x}$$  

$$Iy=\int IQ\ dx$$  

$$y=\left(C_1+C_2x\right)e^{\alpha x}$$  

$$am^2+bm+c=Q$$  

$$am^2+bm+c=0$$  

$$m^2+bm+c=0$$  

$$am^2+bm=c$$  

$$aD^2+bD+c=Q$$  

$$aD^2y+bDy+cy=0^{ }$$  

$$aD^2y+bDy+cy=Q$$  

$$aD^2y+bDy+c=Q$$  

$$\frac{d^2y}{dx^2}$$  

$$\frac{dy}{dx}$$  

$$y''$$  

$$y'$$  

$$\frac{d^2y}{dx^2}$$  

$$\frac{dy}{dx}$$  

$$y''$$  

$$y'$$  

$$y_p=\left(aD^2+bD+c\right)\left(Q\right)$$  

$$y_p=\frac{1}{D}\left(Q\right)$$  

$$y_p=\frac{1}{aD^2+bD+c}\left(Q\right)$$  

$$y_p=\frac{1}{aD^2+bD+c}\left(0\right)$$  

$$y_p=\left(D^2+2D+1\right)\left(5\right)$$  

$$y_p=\frac{1}{D}\left(5\right)$$  

$$y_p=\frac{1}{D^2+2D+1}\left(5\right)$$  

$$y_p=\frac{1}{D^2+2D}\left(5\right)$$  

$$\int f\left(x\right)\ dx$$  

$$\frac{d}{dx}\left(f\left(x\right)\right)$$  

$$f'\left(x\right)$$  

$$\int f'\left(x\right)\ dx$$