Find the particular solution of the following differential equation given that at x = 2, y = 1 $x\,{\frac{d y}{d x}}\,+\,2y=x^{2},\,(x\,\neq\,0).$

Solve the differential equation $x\log x\ {\frac{d y}{d x}}+y\ =$${\frac{2}{x}}\mathrm{log}\,x.$

Solve the differential equation ${\frac{d y}{d x}}\ +\ y\ \cot\ x$= 2 cos x, given that y = 0 when $x={\frac{\pi}{3}}\,.$

Solve the following differential equation : (1 + x²) dy + 2xy dx = cot x dx, (x ≠ 0)

Solve the following differential equation : ${\frac{d y}{d x}}\ +\,\sec x{y}=\tan x,\left(0\leq x<{\frac{\pi}{2}}\right)\!.$

Find the angle of intersection of the curves x² + y² = 4 and (x – 2)² + y²= 4, at the point in the first quadrant.

Solve the differential equation : $(1+x^{2}){\frac{d y}{d x}}\ +y=e^{\tan^{-1}x}.$

Solve the differential equation: $x\,{\frac{d y}{d x}}\ +y=x\cos x+\sin x,$ given $y\left({\frac{\pi}{2}}\right)=1.$

Solve the following differential equation : $(x^{2}-1){\frac{d y}{d x}}~+2x y={\frac{2}{x^{2}-1}},~~x\neq{\bf1}$

Find the equation of tangent to the curve x² + 3y = 3, which is parallel to line y – 4x + 5 = 0.

Find the equations of the normal to the curve y = x³+ 2x + 6, which are parallel to line x + 14y + 4 = 0.

Find the equation of tangent to the curve $y={\frac{x-7}{x^{2}-5x+6}}$ at the point, where it cuts the x-axis.

Find the equation of tangent to the curve 4x² + 9y² = 36 at the point (3 cos θ, 2 sin ).

Find the equation of the tangent to the curve y = x⁴ – 6x³ + 13x² – 10x + 5 at point x = 1, y = 0.

Find the equations of tangents to the curve y = (x²– 1) (x – 2) at the points, where the curve cuts the X-axis.