Solve : $(D^2 - 4D + 1)y = x^2$

> [!TLDR] > Solve the auxiliary equation for the complementary function, then find a polynomial particular integral by the inverse-operator expansion method. > > 1. Given ODE: $(D^2-4D+1)y = x^2$. > 2. **Complementary Function (C.F.):** Auxiliary equation $m^2-4m+1=0$. > $m = \dfrac{4\pm\sqrt{16-4}}{2} = \dfrac{4\pm\sqrt{12}}{2} = 2\pm\sqrt3$. > $\text{C.F.} = C_1 e^{(2+\sqrt3)x} + C_2 e^{(2-\sqrt3)x}$. > 3. **Particular Integral (P.I.):** $\text{P.I.} = \dfrac{1}{D^2-4D+1}x^2 = \dfrac{1}{1-(4D-D^2)}x^2$. > 4. Expand as a binomial series in the operator $u=4D-D^2$ (only terms up to $D^2$ matter, since $D^3x^2=0$): > $\dfrac{1}{1-u} = 1+u+u^2+\cdots$ > 5. Compute $u(x^2) = 4D(x^2)-D^2(x^2) = 4(2x)-2 = 8x-2$. > 6. Compute $u^2(x^2)$: $u^2 = (4D-D^2)^2 = 16D^2-8D^3+D^4$; applied to $x^2$: $16D^2x^2-8D^3x^2+D^4x^2 = 16(2)-0+0=32$. …

Solve : (D² - 4D + 1)y = x² – Class 12 Mathematics

Question 93 of 126

(D^2 - 4D + 1)y = x^2

Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2017Subjective· 6mImportance★★★★★

74% · 93/126 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Second-Order Linear Differential Equations with Constant Coefficients

Second-Order Linear Differential Equations with Constant Coefficients

A linear differential equation with constant coefficients has the form $f(D)\,y = g(x)$ , where $D = \dfrac{d}{dx}$ and $f(D)$ is a polynomial in $D$ , for example $\dfrac{d^2y}{dx^2} + p\dfrac{dy}{dx} + q\,y = g(x)$ . Its general solution is the sum $y = y_c + y_p$ , where the complementary function $y_c$ solves the homogeneous equation $f(D)y = 0$ and the particular integral $y_p$ accounts for $g(x)$ . …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ11 — The integrating factor of the differential equation (dy)/(dx) - ytan x…→Next questionQ13 — A cup of coffee at temperature 100°C is placed in a room whose tempera…

Q.Solve : (D2−4D+1)y=x2(D^2 - 4D + 1)y = x^2(D2−4D+1)y=x2

Concept understanding — Second-Order Linear Differential Equations with Constant Coefficients

Second-Order Linear Differential Equations with Constant Coefficients

🎓Unlock everything free for 14 days

Q.Solve : $(D^2 - 4D + 1)y = x^2$

Unlock everything free for 14 days