1. Solve differential equations by the method of variation of parameters. A. Given: $$\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = e^x \sin x$$ - Find complementary function (C.F.) by solving homogeneous equation: $$\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = 0$$ - Characteristic equation: $$r^2 - 2r = 0 \Rightarrow r(r-2)=0 \Rightarrow r=0, 2$$ - C.F. is $$y_c = C_1 + C_2 e^{2x}$$ - Use variation of parameters to find particular solution (P.S.) for nonhomogeneous part. - Final solution given: $$y = C_1 + C_2 e^{2x} - \frac{e^x}{2} \sin 2x$$ B. Given: $$\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = e^x \ln x$$ - Solve homogeneous: characteristic $$r^2 - 2r + 1 =0 \Rightarrow (r-1)^2=0 \Rightarrow r=1$$ repeated root. - C.F.: $$y_c = (C_1 + C_2 x) e^x$$ - Variation of parameters for P.S. - Given solution: $$y = (C_1 + C_2 x) e^x + x^2 e^x \frac{2(\ln x - 3)}{4}$$ C. Problem: $$(\theta^2 -1) y = \frac{2}{(1 - e^{-2x})^{1/2}}$$ - C.F. typical form: $$y_c = C_1 e^x + C_2 e^{-x}$$ - P.S. involves terms with $$e^x \sin (\theta^2 x)$$ and exponential expressions. - Final solution: $$y = C_1 e^x + C_2 e^{-x} - e^x \sin(\theta^2 x) - e^x (e^{2x} -1)$$ D. Cauchy Euler equation: $$x^2 y'' + x y' - y = x^2 e^x$$ - Find C.F. solving $$x^2 y'' + x y' - y =0$$ - C.F. is $$y_c = C_1 x + \frac{5}{x}$$ - P.S. from method of variation of parameters or undetermined coefficients - Final: $$y = C_1 x + \frac{5}{x} + \left(1 - \frac{1}{x}\right) e^x$$ E. Another Cauchy Euler: $$x^2 y'' + 2x y' - 12y = x^3 \ln x$$ - C.F. by characteristic equation. - Final solution: $$y = x^2 \left[\frac{1}{4} (\ln x)^2 + a x^3 + \frac{x^2}{4.9}\left(7/7 - \ln x\right) + f_2 x^3 \right]$$ 2. Solve DEs with differential operators. A. $$(D^4 + 6D^3 +11 D^2 + 6D) y = 20 e^{-2x} \sin x$$ B. $$(D^3 - 4D + 4)y = 8 x^2 e^{2x} \sin 2x$$ C. $$(D^2 + 2D +1) y = x \cos x$$ D. $$(D^2 + 4D -12)y = (x-1) e^{2x}$$ E. $$(D^2 - 2D +1) y = x e^{x} \cos x$$ 3. Solve DEs with third derivatives and variable coefficients. A. $$x^2 \frac{d^3 y}{d x^3} - 4x \frac{d^2 y}{d x^2} + 6 \frac{dy}{dx} = 4$$ - Solution: $$y = C_1 + C_2 x^3 + C_3 x^4 + \frac{2}{3} x$$ B. $$x^4 \frac{d^3 y}{d x^3} + 2 x^3 \frac{d^2 y}{d x^2} - x^2 \frac{dy}{dx} - x y = 1$$ - Solution: $$y = (C_1 + C_2 \ln x) x + \frac{C_2}{x} + \frac{1}{4} x \ln x$$ C. With substitution \(2x+3 = e^z\): $$y = C_1 (2x+3)^{-1} + C_2 (2x+3)^{-3/2} (2x+3 + 3)$$ D. $$\frac{d^3 y}{dx^3} - \frac{4}{x} \frac{d^2 y}{dx^2} + \frac{5}{x^2} \frac{dy}{dx} - \frac{2}{x^3} y = 1$$ - Solution: $$y = C_1 x^2 + \frac{x^3}{2} \left(C_2 x^{\sqrt{21}/2} + C_3 x^{-\sqrt{21}/2} \right) - \frac{x^3}{5}$$