Embibe Experts Solutions for Exercise 4: Assignment Set-1

It is known that, if the interest is compounded continuously, the principal changes at the rate equal to the product of the rate of bank interest per annum and the principal.

How much will $\mathrm{Rs}. 1000$ be worth at $5\%$  interest after $10$ years? $\sqrt{e}=1.648$.

Verify that

$\left(x^2-y^2\right)=c{\left(x^2+y^2\right)}^2$ satisfies the differential equation $\left(x^3-3x{y}^2\right)dx=\left(y^3-3x^{2}y\right)dy$

A radioactive substance disintegrates at a rate proportional to the amount of substance present. If $50\%$ of the given amount disintegrates in $1600$ years. What percentage of the substance disintegrates in $10$ years?

Find the curve in $x-y$ plane passing through the point $(1,1)$ so that the segment of any tangent drawn to the curve between its point of tangency and the $y$-axis is bisected at the $x$-axis.

Solve the differential equation $\left(x^3+x^2+x+1\right)\frac{dy}{dx}=2x^2+x,y=1$ when $x=0$.

Form a differential equation representing the given family of curve by eliminating arbitrary constants $a$ and $b$.

$y=a{e}^{2x}+bx{e}^{2x}$

Find the curve through $(2,0)$ so that the segment of tangent between point of tangency and $y$-axis has a constant length equal to $2$ .

Find the equation of a curve passing through origin if the midpoint of the segment of the normal drawn at any point of the curve and the $x$-axis lies on the parabola $2y^2=x$.