Related Formula
Standard form LDE: (dy)/(dx) + P(x)y = Q(x)$$\text{Standard form LDE: } \frac{dy}{dx} + P(x)y = Q(x)$$
Integrating Factor (I.F.) = e∫ P(x)dx$$\text{Integrating Factor (I.F.) } = e^{\int P(x)dx}$$
Solution is y(I.F.) = ∫ Q(x)(I.F.)dx + C$$\text{Solution is } y(I.F.) = \int Q(x)(I.F.)dx + C$$
Core Logic
Multiply the entire differential equation by x$\cos x$ to normalize the (dy)/(dx)$\frac{dy}{dx}$ term:
(dy)/(dx) - (2 x)y = 2 x + 3 x x$$\frac{dy}{dx} - (2\cos x)y = 2\cos x + 3\sin x\cos x$$
This is a standard first-order linear differential equation with P(x) = -2 x$P(x) = -2\cos x$.
Step 1: Integrating Factor
I.F. = e∫ -2 x dx = e-2 x$$\text{I.F.} = e^{\int -2\cos x \, dx} = e^{-2\sin x}$$
Step 2: General Solution
y · e-2 x = ∫ e-2 x (2 x + 3 x x) dx$$y \cdot e^{-2\sin x} = \int e^{-2\sin x} (2\cos x + 3\sin x\cos x) \, dx$$
Let u = x du = x dx$u = \sin x \implies du = \cos x \, dx$.
The integral becomes:
∫ e-2u (2 + 3u) du$$\int e^{-2u} (2 + 3u) \, du$$
Using integration by parts:
= (2+3u)( e-2u-2) - ∫ 3 ( e-2u-2) du$$= (2+3u)\left(\frac{e^{-2u}}{-2}\right) - \int 3 \left(\frac{e^{-2u}}{-2}\right) \, du$$
= -(2+3u)/(2) e-2u - (3)/(4) e-2u$$= -\frac{2+3u}{2} e^{-2u} - \frac{3}{4} e^{-2u}$$
= e-2u (-(2+3u)/(2) - (3)/(4)) = e-2u (-(4+6u+3)/(4)) = e-2u (-(6u+7)/(4))$$= e^{-2u} \left(-\frac{2+3u}{2} - \frac{3}{4}\right) = e^{-2u} \left(-\frac{4+6u+3}{4}\right) = e^{-2u} \left(-\frac{6u+7}{4}\right)$$
Re-substitute u = x$u = \sin x$:
y e-2 x = e-2 x (-(3)/(2) x - (7)/(4)) + C$$y e^{-2\sin x} = e^{-2\sin x} \left(-\frac{3}{2}\sin x - \frac{7}{4}\right) + C$$
y = -(3)/(2) x - (7)/(4) + C e2 x$$y = -\frac{3}{2}\sin x - \frac{7}{4} + C e^{2\sin x}$$
Step 3: Apply Boundary Conditions
Given y(0) = -(7)/(4)$y(0) = -\frac{7}{4}$:
-(7)/(4) = -(3)/(2)(0) - (7)/(4) + C e⁰ C = 0$$-\frac{7}{4} = -\frac{3}{2}(0) - \frac{7}{4} + C e^0 \implies C = 0$$
Thus, y(x) = -(3)/(2) x - (7)/(4)$y(x) = -\frac{3}{2}\sin x - \frac{7}{4}$.
Step 4: Find Final Value
Calculate y((π)/(6))$y\left(\frac{\pi}{6}\right)$:
y((π)/(6)) = -(3)/(2) ((π)/(6)) - (7)/(4) = -(3)/(2)((1)/(2)) - (7)/(4)$$y\left(\frac{\pi}{6}\right) = -\frac{3}{2}\sin\left(\frac{\pi}{6}\right) - \frac{7}{4} = -\frac{3}{2}\left(\frac{1}{2}\right) - \frac{7}{4}$$
= -(3)/(4) - (7)/(4) = -(10)/(4) = -(5)/(2)$$= -\frac{3}{4} - \frac{7}{4} = -\frac{10}{4} = -\frac{5}{2}$$
Pattern Recognition
When integrating eax(f(x))$e^{ax}(f(x))$, use DI method or standard Integration by Parts mapping. For ∫ e-2u(3u+2)$\int e^{-2u}(3u+2)$, always extract polynomial as u$u$, exponential as dv$dv$.
Chapter Mix
Class 12 Maths: Differential Equations
