Equations Reducible to Standard Forms

Q. If y = √x÷a + √a÷x , prove that 2xy dy/dx = (x/a - a/x) .

Q. Prove The image of z1= -15-16i with respect to |z-1-i|=4 as 17+18i.

Q. The general solution of the differential equation $(y^2–x^3)dx–xydy=0(x\ne 0)$ is : (where $c$ is a constant of integration)

1. $y^2+2x^2+c{x}^3=0$
2. $y^2-2x^3+c{x}^2=0$
3. $y^2+2x^3+c{x}^2=0$
4. $y^2–2x^2+c{x}^3=0$

Q. $\mathrm{The}\mathrm{solution}\mathrm{of}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{x+y+1}{2x+2y+3}\mathrm{is}$.

1. $\frac{2}{3}\left(x+y\right)+\frac{1}{9}\log \left(2x+2y+4\right|=x+c$
2. $\frac{2}{3}\left(x+y\right)+\frac{1}{9}\log \left(3x+3y+4\right|=x+c$
3. $\frac{2}{3}\left(x+y\right)+\frac{1}{9}\log \left(3x+3y\right|=x+c$
4. $\frac{1}{9}\log \left(3x+3y+4\right|=x+c$

Q. Find dy/dx if y=(ax^3 + 3)^4

Q. Solve the followings: (1) ʃcosec2xcot2xdx = ? (2) ʃ[e^(5logx)]dx = ? (3) ʃsinx°dx = ?

Q.

Let f(x) is a continuous function which takes positive values for x (x>0), and satisfy ${\int }_0^{x}f\left(t\right)dt=x\sqrt{f\left(x\right)}$ with $f\left(1\right)=\frac{1}{2}$. Then the value of $f(\sqrt{2}+1)$ equals

1. 1
2. $\sqrt{2}-1$
3. $\frac{1}{4}$
4. $\frac{1}{\sqrt{2}-1}$