Differential Equations - Class 12 Commerce Applied Mathematics - Extra Questions

If $$y=Ae^{mx}+Be^{nx}$$, show that $$\dfrac{d^2y}{dx^2}-(m+n)\dfrac{dy}{dx}+mny=0$$.



Solve: $$\left( {1 + {y^2}} \right)dx + \left( {x - {e^{ - {{\tan }^{ - 1}}y}}} \right)dy = 0,y\left( 0 \right) = 0$$



Integrating factor of $$\cfrac{xdy}{dx}-y={x}^{4}-3x$$ is:



The volume of a sphere is increasing at the rate of 3 cubic centimetre per second. Find the rate of its surface area, when the radius is 2cm.



The differential equation of parabola with axis parallel to y-axis and length of latus rectum as 4a is of order:



Find the integrating factor of the differential equation $$\left(\dfrac{e^{-2\sqrt{x}}}{\sqrt{x}}-\dfrac{y}{\sqrt{x}}\right)\dfrac{dx}{dy}=1$$.



The rate at which the population of a city increases at any time is proportional to the population at that time. If there are $$1,30,000$$ people in the city in $$1960$$ and $$1,60,000$$ in $$1990$$, what approximate population may be anticipated in $$2020$$?
$$\left[ \log _{ e }{ \cfrac { 16 }{ 13 }  } =0.2070,{ e }^{ 0.42 }=1.52 \right] $$



If the population of a country doubles in $$60$$ years, in how many years will it be triple under the assumption that the rate of increase is proportional to the number of inhabitants?
[Given: $$\log 2 = 0.6912$$ and $$\log 3 = 1.0986$$.]



The solution of $$\dfrac{{dy}}{{dx}} = \dfrac{{x\log {x^2} + x}}{{\sin y + y\cos y}}$$ is 



Solve $$2y\,\,{e^{\dfrac{x}{y}}}\,\,dx\, + \,\left( {y - 2x\,{e^{\dfrac{x}{y}}}} \right)\,\,dy = 0$$



Solve $$\dfrac{dy}{dx}=xcosx$$



The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was $$200000$$ in $$1990$$ and $$250000$$ in $$2000$$, what will be the population in $$2010$$?



Represent the following families of curves by forming the corresponding differential equation.(a, b being parameters).
$$y^2=4ax$$.



The population of a city increases at a rate proportional to the number of inhabitants present at any time t. If the population of the city was $$200000$$ in $$1990$$ and $$250000$$ in $$2000$$, what will be the population in $$2010$$?



Let $$\mathrm{f}:\mathrm{R}\rightarrow \mathrm{R}$$ be a continuous function which satisfies $$\displaystyle \mathrm{f}(\mathrm{x})=\int_{0}^{\mathrm{x}}\mathrm{f}(t)dt$$ Then the value of $$\mathrm {f(ln5)}$$ is



A function $$y = f(x)  $$  satisfying  $$    f"(x) = x^{-3/2}, f'(4) = 2  $$  and $$ f(0) = 0 $$ is $$ -\lambda \sqrt{x}+3x$$,  then find the value of  $$\lambda$$



The largest value of $$\dfrac{2}{\pi}c$$ such that there exists a differentiable function $$h(x)$$ for $$-c <x <c$$ that is a solution of $$y'=1+y^{2}$$ with $$h(0) = 0$$ is:



A tank consists of 50 liters of fresh water. Two liters of brine each litre containing 5 gms of dissolved salt are running into tank per minute ; the mixture is kept uniform by stirring , and runs out at the rate of one litre per minute.If $$m$$ grams of salt are present in the tank after t minute. expression of $$m$$ in terms of $$t$$ be$$\quad m\, =\, kt \left ( r+\displaystyle \frac {2p+t}{p+t} \right ) gm$$. Find $$(p/k)+r$$ ?



Let $$(x^2 + 1) \displaystyle \frac{d^2 y}{dx^2} = 2x \frac{dy}{dx},$$ where $$y'(0) = 3$$ & $$y(0) = 1$$, then $$y(1)$$ is equal to:



Find the particular solution of the following differential equation;
$$(x + 1)\dfrac {dy}{x} = 2e^{-y} - 1; y = 0$$ when $$x = 0$$



Find the particular solution of the following differential equation:
$$(x+1)\dfrac{dy}{dx} = 2e^{-y} - 1$$, given that $$y = 0$$ when $$ x = 0$$.



The temperature $$T$$ of a cooling object drops at a rate proportional to the difference $$(T - S)$$, where $$S$$ is constant temperature of surrounding medium. If initially $$T = 150^{\circ}C$$, find the temperature of the cooling object at any time $$'t'$$.



The sum of Rs. $$1000$$ is compounded continuously, the nominal rate of interest being four percent per annum. In how many years will the amount be twice the original principal? $$(\log_e2 = 0.6931)$$.



A radioactive substance disintegrates at a rate proportional to its mass. When its mass is $$10 $$ mg, the rate of disintegration is $$0.051 $$ mg per day. How long will it take for the mass to be reduced from $$10$$ mg to $$5$$ mg? $$\left( \log _{ 2 }{ 2 } =0.6931 \right) $$.



A body is heated at $$110^{\circ}C$$ and placed in air at $$10^{\circ}C$$. After $$1$$ hour its temperature is $$60^{\circ}C$$. How much additional time is required for it to cool to $$35^{\circ}C$$?



The population of a town grows at the rate of $$10\%$$ per year. Using differential equation, find how long will it take for the population to grow $$4$$ times.



The rate of growth of population is proportional to the number present. If the population doubled in the last $$25$$ years and the present population is $$1$$ lac, when will the city have population $$4,00,000$$ ?



$$(x^{2} + y^{2}) dy = xy dx$$. If $$y(1)$$ and $$y(x_{0})= e$$ then find the value of $$x_{0}$$.



Class 12 Commerce Applied Mathematics Extra Questions