Find the general solution of the differential equation : $(x-y){\frac{d y}{d x}}\,=x+2y.$

Solve the following differential equation : ${\frac{d y}{d x}}={\frac{y}{x}}+{\frac{\sqrt{x^{2}+y^{2}}}{x}},\,x>0\,.$

Solve the differential equation : $y+x{\frac{d y}{d x}}=x-y{\frac{d y}{d x}}.$

Solve the following differential equation : $\left(1+e^{\frac{x}{y}}\right)d x+e^{\frac{x}{y}}\left(1-{\frac{x}{y}}\right)d y=0.$

Find the solution of the differential equation $(x d y-y d x)\,y\,\mathrm{sin}{\left({\frac{y}{x}}\right)}=(y d x+x d y)x\cos\biggl({\frac{y}{x}}\biggr).$

Find the general solution of the differential equation : $x d y-y d x={\sqrt{x^{2}+y^{2}}}d x~.$

Find the particular solution of the differential equation $2y e^{x/y}d x+(y-2x e^{x/y})d y=0$ given that x = 0 when y = 1.

Find the approximate value of f(3·02), up to 2 places of decimal, where f(x) = 3x² + 5x + 3.

Find the general solution of the differential equation: ${\frac{d x}{d y}}\;=\;{\frac{y\tan y-x\tan y-x y}{y\tan y}}$

If the function f : R → R is given by f(x) = x² + 3x + 1 and g : R → R is given by g(x) = 2x – 3, then find (i) fog (ii) gof

If the radius of sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.

Using differentials, find the approximate value of $(3.968)^{1/2}$

Using differentials, find the approximate value of ${\sqrt{49\cdot5}}\ .$

If the function f : R → R be defined by f(x) = 2x – 3 and g : R → R by g(x) = x³ + 5, then find fog and show that fog is invertible. Also, find (fog)⁻¹, hence find (fog)⁻¹(9).

If the function f : R → R is given by f(x) = $\frac{x+3}{3}$ and g : R → R is given by g(x) = 2x – 3, then find (i) fog (ii) gof Is f⁻¹ = g?