1. **Problem Q1A:** Solve $$\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = e^x \ln x$$ by variation of parameters. - The complementary equation is $$\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} = 0$$ with characteristic equation $$m^2 - 2m = 0$$, roots $$m=0, 2$$. So complementary solution: $$y_c = C_1 + C_2 e^{2x}$$. - Using variation of parameters, identify $$y_1=1$$ and $$y_2=e^{2x}$$. 2. **Problem Q1B:** Solve $$\frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + y = e^x \ln x$$. - Complementary equation roots of $$m^2 - 2m +1=0$$ are $$m=1$$ repeated root. So $$y_c = (C_1 + C_2 x) e^x$$. - Apply variation of parameters accordingly. 3. **Problem Q1C:** Solve $$(D^2 - 1)y = \frac{2}{\sqrt{1 - e^{-2x}}}$$. - Complementary roots from $$m^2 - 1=0$$ are $$m=\pm1$$, so $$y_c=C_1 e^{x} + C_2 e^{-x}$$. - Use variation of parameters with given RHS. 4. **Problem Q1D:** Cauchy-Euler equation $$x^2 y'' + x y' - y = x^2 e^{x}$$. - Characteristic equation for Cauchy-Euler: $$m^2 -1 = 0 \Rightarrow m=\pm 1$$, so $$y_c=C_1 x + C_2 / x$$. - Use method of variation of parameters or undetermined coefficients for particular solution. 5. **Problem Q1E:** Cauchy-Euler equation $$x^2 y'' + 2x y' - 12 y = x^3 \log x$$. - Characteristic: $$m^2 + m -12=0$$ root $$m=3,-4$$. Complementary solution: $$y_c=C_1 x^3 + C_2 x^{-4}$$. - Using variation or other techniques to find particular solution with logarithm. 6. **Problem Q2A:** Solve $$x^2 y''' - 4 x y'' + 6 y' = 4$$ with given solution. Confirm solution via substitution. 7. **Problem Q2B:** Solve $$x^4 y''' + 2 x^3 y'' - x^2 y' - x y = 1$$. Substitute given solution to verify. 8. **Problem Q2C:** Let $$z = \log(2x+3)$$, transform the DE $$ (2x+3)^2 y'' - 2 (2x+3) y' - 12 y = 6 x$$ to a simpler form in $$z$$. Use substitution $$D_1 = \frac{d}{dz}$$ and solve for $$y$$. 9. **Problem Q2D:** Solve $$y''' - 4 y'' + \frac{5}{x^2} y' - \frac{2}{x^3} y = 1$$, verify provided solution with powers and constants. 10. **Problem Q3A to Q3E:** These involve linear differential operators $$D$$ with varying powers and forcing functions involving exponentials and trigonometric functions. The typical approach is: - Find the characteristic polynomial roots. - Write complementary solution. - Use method of undetermined coefficients or variation of parameters for the particular integrals involving exponential sinusoidal inputs. --- **Final summary:** This set covers different methods of solving linear differential equations including Cauchy-Euler equations, variation of parameters, and handling polynomial/exponential/trigonometric forcing terms. Stepwise solving involves: - Finding characteristic roots, writing complementary solutions - Applying variation of parameters formulas stepwise - Verifying with given answers by substitution Each problem counts as separate question, yielding q_count: 14.