1.Solve the differential equation $\frac{dy}{dx} + y = e^{-x}$ with initial condition $y(0) = 1$.
Sol: Integrating factor $I.F. = e^{\int dx} = e^x$. $y e^x = \int e^x e^{-x} dx = x + C \implies y = (x + C)e^{-x}$. At $x=0, y=1 \implies C=1$. Solution: $y = (x + 1)e^{-x}$.
2.Evaluate the Gaussian integral $\int_{-\infty}^{\infty} e^{-x^2} dx$.
Sol: $I^2 = \int_{-\infty}^{\infty} e^{-x^2} dx \int_{-\infty}^{\infty} e^{-y^2} dy = \int_0^{2\pi} d\theta \int_0^\infty e^{-r^2} r dr = 2\pi \cdot \frac{1}{2} = \pi \implies I = \sqrt{\pi}$.
3.Find the area bounded by the curve $y = \sin x$ and the x-axis from $x = 0$ to $x = 2\pi$.
Sol: Total physical area = $\int_0^\pi \sin x dx + \left|\int_\pi^{2\pi} \sin x dx\right| = 2 + 2 = 4\text{ sq units}$.
4.A particle moves along a curve $r(\theta) = a(1 + \cos\theta)$. Find radial and transverse components of velocity.
Sol: Radial velocity $v_r = \dot{r} = -a \sin\theta \dot{\theta}$. Transverse velocity $v_\theta = r \dot{\theta} = a(1 + \cos\theta)\dot{\theta}$.
