$~16.~~$ In a certain culture the number of bacteria at any instant increases at a rate proportional to the cube root…

$~8.~~$ A curve passes through the point $~(2,0)~$ and satisfies the equation $~y~dy +4~dx=0;$ show that the curve is a…

$~1.~$ If $~f'(x)=xe^x~$ and $~f(0)=1,~$ find $~f(x).$ Sol. $~~f'(x)=xe^x \\ \therefore f(x)=\displaystyle\int{…

Solve:  $~7.~~\frac{dy}{dx}=\frac{x-y+1}{2x-2y+3}$ Sol. $~~~\frac{dy}{dx}=\frac{x-y+1}{2x-2y+3} \\ \text{or,}~~ \frac{d…

Solve :  $~1.~~\frac{dy}{dx}=\frac{4x-5y+3}{5x+y-2}$ Sol. $~~\frac{dy}{dx}=\frac{4x-5y+3}{5x+y-2}\rightarrow(1)$ Let $~…

Find the particular solutions of the following equations :   $~26.~~\frac{dy}{dx}=\frac{3x+2y}{2x-3y};\\~~\text{given}~…

$~20.~~x~dy-y~dx=\sqrt{x^2+y^2}~dx$ Sol. $~~~x~dy-y~dx=\sqrt{x^2+y^2}~dx \\ \text{or,}~~x~dy=(\sqrt{x^2+y^2}+y)~dx \\…

$~13.~~\frac{dy}{dx}=\frac yx+\cot\frac yx$ Sol. $~~~\frac{dy}{dx}=\frac yx+\cot\frac yx \rightarrow(1)$ Let $~~~~y=v…

$~7.~~(x^2+xy)~\frac{dy}{dx}=x^2+y^2$ Sol. $~~(x^2+xy)~\frac{dy}{dx}=x^2+y^2 \\ \Rightarrow \frac{dy}{dx}=\frac{x^2+y…

$~1.~~\frac{dy}{dx}=\frac{y-x}{x+y}$ Sol. $~~~\frac{dy}{dx}=\frac{y-x}{x+y}\rightarrow(1)$ Let $~~~~y=vx \\ \therefor…