Class 12

Math

Calculus

Differential Equations

In a bank, principal increases continuously at the rate of 5% per year. An amountof Rs 1000 is deposited with this bank, how much will it worth after 10 years$(e_{0.5}=1.648)$

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The general solution of the differential equation, $y_{prime}+yϕ_{prime}(x)−ϕ(x)ϕ_{prime}(x)=0$ , where $ϕ(x)$ is a known function, is

The force of resistance encountered by water on a motor boat of mass $m$ going in still water with velocity $v$ is proportional to the velocity $v˙$ At $t=0$ when its velocity is $v_{0},$ then engine shuts off. Find an expression for the position of motor boat at time $t$ and also the distance travelled by the boat before it comes to rest. Take the proportionality constant as $k>0.$

The solution of the differential equation $dx(x+2y_{3})dy =y$ is

The slope of the tangent at $(x,y)$ to a curve passing through $(1,4π )$ is given by $xy −cos_{2}(xy ),$ then the equation of the curve is

A curve $y=f(x)$ passes through $(1,1)$ and tangent at $P(x,y)$ cuts the x-axis and y-axis at $A$ and $B$ , respectively, such that $BP:AP=3,$ then (a) equation of curve is $xy_{prime}−3y=0$ (b) normal at $(1,1)$ is $x+3y=4$ (c) curve passes through $2,8$ (d) equation of curve is $xy_{prime}+3y=0$

Let $f:[1,∞]$ be a differentiable function such that $f(1)=2.$ If $6∫_{1}f(t)dt=3xf(x)−x_{3}$ for all $x≥1,$ then the value of $f(2)$ is

From the differential equation of family of lines situated at a constant distance p from the origin.

If $∫_{a}ty(t)dt=x_{2}+y(x),$ then find $y(x)$