Differential Equations · Mathematics · MHT CET

If the slope of the tangent of the curve at any point is equal to $$-y+\mathrm{e}^{-x}$$, then the equation of the curve passing through origin is...

If a body cools from $$80^{\circ} \mathrm{C}$$ to $$50^{\circ} \mathrm{C}$$ in the room temperature of $$25^{\circ} \mathrm{C}$$ in 30 minutes, then t...

The differential equation representing the family of curves $$y^2=2 \mathrm{c}(x+\sqrt{\mathrm{c}})$$, where $$\mathrm{c}$$ is a positive parameter, i...

General solution of the differential equation $$\cos x(1+\cos y) \mathrm{d} x-\sin y(1+\sin x) \mathrm{d} y=0$$ is

The differential equation of $$y=\mathrm{e}^x(\mathrm{a} \cos x+\mathrm{b} \sin x)$$ is

If $$x d y=y(d x+y d y), y(1)=1, y(x)>0$$, then $$y(-3)$$ is

The solution of $$(1+x y) y d x+(1-x y) x d y=0$$ is

A radioactive substance, with initial mass $$m_0$$, has a half-life of $$h$$ days. Then, its initial decay rate is given by

The solution of the differential equation $$\mathrm{e}^{-x}(y+1) \mathrm{d} y+\left(\cos ^2 x-\sin 2 x\right) y \mathrm{~d} x=0$$ at $$x=0$$, $$y=1$$ ...

Rate of increase of bacteria in a culture is proportional to the number of bacteria present at that instant and it is found that the number doubles in...

The particular solution of differential equation $$\mathrm{e}^{\frac{d y}{d x}}=(x+1), y(0)=3$$ is

A right circular cone has height $$9 \mathrm{~cm}$$ and radius of base $$5 \mathrm{~cm}$$. It is inverted and water is poured into it. If at any insta...

The solution of $$\mathrm{e}^{y-x} \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y(\sin x+\cos x)}{(1+y \log y)}$$ is

Water flows from the base of rectangular tank, of depth 16 meters. The rate of flow of the water is proportional to the square root of depth at any ti...

If $$(2+\sin x) \frac{\mathrm{d} y}{\mathrm{~d} x}+(y+1) \cos x=0$$ and $$y(0)=1$$, then $$y\left(\frac{\pi}{2}\right)$$ is

The decay rate of radio active material at any time $$t$$ is proportional to its mass at that time. The mass is 27 grams when $$t=0$$. After three hou...

The differential equation $$\cos (x+y) \mathrm{d} y=\mathrm{d} x$$ has the general solution given by

If $$\frac{\mathrm{d} y}{\mathrm{~d} x}=y+3$$ and $$y(0)=2$$, then $$y(\log 2)=$$

The solution of $$\frac{\mathrm{d} x}{\mathrm{~d} y}+\frac{x}{y}=x^2$$ is

The solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{y}{x}=\sin x$$ is

The solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1+y^2}{1+x^2}$$ is

A curve passes through the point $$\left(1, \frac{\pi}{6}\right)$$. Let the slope of the curve at each point $$(x, y)$$ be $$\frac{y}{x}+\sec \left(\f...

The general solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{3 x^2}{1+x^3}\right) y=\frac{1}{x^3+1}$$ is

The differential equation of all circles, passing through the origin and having their centres on the $$\mathrm{X}$$-axis, is

The population $$\mathrm{P}=\mathrm{P}(\mathrm{t})$$ at time $$\mathrm{t}$$ of certain species follows the differential equation $$\frac{d P}{d t}=0.5...

The differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sqrt{1-y^2}}{y}$$ determines a family of circles with

General solution of the differential equation $$\cos x(1+\cos y) \mathrm{d} x-\sin y(1+\sin x) \mathrm{d} y=0$$ is

General solution of the differential equation $$\log \left(\frac{d y}{d x}\right)=a x+b y$$ is

The differential equation of all circles which pass through the origin and whose centres lie on $$\mathrm{Y}$$-axis is

The differential equation of all parabolas, whose axes are parallel to $$\mathrm{Y}$$-axis, is

The particular solution of the differential equation $$\left(1+y^2\right) \mathrm{d} x-x y \mathrm{~d} y=0$$ at $$x=1, y=0$$, represents

The general solution of the differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3 x+y}{x-y}$$ is (where $$C$$ is a constant of integratio...

The differential equation $$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sqrt{1-y^2}}{y}$$ determines a family of circles with

The particular solution of the differential equation $$\left(1+e^{2 x}\right) d y+e^x\left(1+y^2\right) d x=0$$ at $$x=0$$ and y = 1 is

The order and degree of the differential equation $$\sqrt{\frac{d y}{d x}}-4 \frac{d y}{d x}-7 x=0$$ are respectively.

A population P grew at the rate given by the equation $$\frac{dP}{dt}=0.5P$$, then the population will become double in

The differential equation of all parabolas whose axis is $$y$$-axis, is

The general solution of the differential equation $$\frac{d y}{d x}=\tan \left(\frac{y}{x}\right)+\frac{y}{x}$$ is

The particular solution of the differential equation $$ \frac{d y}{d x}=\frac{x+y+1}{x+y-1} $$ when $$ \mathrm{x}=\frac{2}{3} $$ and $$ y=\frac{1}{3} ...

The order of the differential equation whose solution is $$y=a \cos x+b \sin x+c e^{-x}$$ is

The general solution of the differential equation $$(2 y-1) d x-(2 x+3) d y=0$$ is

The differential equation of the family of parabolas with focus at the origin and the $$X$$-axis a axis, is

Radium decomposes at the rate proportional to the amount present at any time. If $$\mathrm{P} \%$$ of amount disappears in one year, then amount of ra...

The differential equation obtained by eliminating A and B from $$y=A \cos \omega t+B \sin \omega t$$

The particular solution of the differential equation $$y(1+\log x) \frac{d x}{d y}-x \log x=0$$ when $$x=e, y=e^2$$ is

The order and degree of the differential equation $$\frac{d^2 y}{d x^2}=\sqrt{\frac{d y}{d x}}$$ are respectively

The general solution of the differential equation $$\cos (x+y) \frac{d y}{d x}=1$$ is

The general solution of the differential equation $$\frac{d y}{d x}+\frac{y^2+y+1}{x^2+x+1}=0$$ is

The general solution of the differential equation $$\frac{d x}{d t}=\frac{x \log x}{t}$$ is

The particular solution of differential equation $$(x+y) d y+(x-y) d x=0$$ at $$x=y=1$$ is

The general solution of the differential equation $$\frac{d y}{d x}=2^{y-x}$$ is

If the surrounding air is kept at $$25^{\circ} \mathrm{C}$$ and a body cools from $$80^{\circ} \mathrm{C}$$ to $$50^{\circ} \mathrm{C}$$ in 30 minutes...

The degree of the differential equation whose solution is $$y^2=8 a(x+a)$$, is

A spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is 3 mm. and 1 hour later has been reduced to 2 m...

The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is

The particular solution of the differential equation $$\frac{d y}{d x}=\frac{y+1}{x^2-x}$$, when $$x=2$$ and $$y=1$$ is

The general solution of $$\frac{d y}{d x}=\frac{x+y}{x-y}$$ is

The particular solution of the diffrential equation $$y(1+\log x)=\left(\log x^x\right) \frac{d y}{d x}$$, when $$y(e)=e^2$$ is

The general solution of $$\sin ^{-1}\left(\frac{d y}{d x}\right)=x+y$$ is

Solution of the differential equation $$\mathrm{y'=\frac{(x^2+y^2)}{xy}}$$, where y(1) = $$-$$2 is given by

The differential equation of all family of lines $$y=m x+\frac{4}{m}$$ obtained by eliminating the arbitrary constant $$\mathrm{m}$$ is

$$\text{I} : y^{\prime}=\frac{y+x}{x} ; \quad \text { II }: y^{\prime}=\frac{x^2+y}{x^3} ; \quad \text { III }: y^{\prime}=\frac{2 x y}{y^2-x^2}$$ S1 ...

The differential equation of the family of circles touching $$y$$-axis at the origin is

The general solution of the differential equation. $$\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right) d x-\left[\left(\frac{x}{y}\right) \sin \l...

If the half life period of a substance is 5 years, then the total amount of the substance left after 15 years, when initial amount is 64 gms is

If $$m$$ is order and $$n$$ is degree of the differential equation $$\left(\frac{d^2 y}{d x^2}\right)^5+4 \frac{\left(\frac{d^2 y}{d x^2}\right)}{\lef...

The general solution of the differential equation $$\left(3 x y+y^2\right) d x+\left(x^2+x y\right) d y=0$$ is

The differential equation of family of circles whose centres lie on $$\mathrm{X}$$-axis is

The general solution of the differential equation $$y(1+\log x)\left(\frac{d x}{d y}\right)-x \log x=0$$ is

The general solution of the differential equation $$\frac{d y}{d x}=\frac{x+2 y-1}{x+2 y+1}$$ is

If $$\mathrm{m}$$ is order and $$\mathrm{n}$$ is degree of the differential equation $$y=\frac{d p}{d x}+\sqrt{a^2 p^2-b^2}$$, where $$p=\frac{d y}{d ...

The general solution of the differential equation $$\cos x \cdot \sin y d x+\sin x \cdot \cos y d y=0$$ is

The differential equation of an ellipse whose major axis is twice its minor axis, is

The general solution of $$\left(x \frac{d y}{d x}-y\right) \sin \frac{y}{x}=x^3 e^x$$ is

The population of a city increases at a rate proportional to the population at that time. If the population of the city increase from 20 lakhs to 40 l...

A differential equation for the temperature 'T' of a hot body as a function of time, when it is placed in a bath which is held at a constant temperatu...

The general solution of the differential equation $$\frac{d y}{d x}=\frac{x+y+1}{x+y-1}$$ is given by

The general solution of the differential equation $$x+y \frac{d y}{d x}=\sec \left(x^2+y^2\right)$$ is

The differential equation of all circles which pass through the origin and whose centre lie on Y-axis is

An ice ball melts at the rate which is proportional to the amount of ice at that instant. Half the quantity of ice melts in 20 minutes, $$x_0$$ is the...

The integrating factor of the differential equation $x \frac{d y}{d x}+y \log x=x^2$ is

The rate of disintegration of a radio active element at time $t$ is proportional to its mass, at the time. Then the time during which the original mas...

The general solution of the differential equation $\left(1+y^2\right)+\left(x-e^{\tan ^{-1} y}\right) \frac{d y}{d x}=0$ is

The order and degree of the differential equation $\left[1+\frac{1}{\left(\frac{d y}{d x}\right)^2}\right]^{\frac{5}{3}}=5 \frac{d^2 y}{d x^2}$ are re...

If the population grows at the rate of $8 \%$ per year, then the time taken for the population to be doubled, is (Given $\log 2=0.6912$)

The integrating factor of the differential equation $$\sin y\left(\frac{d y}{d x}\right)=\cos y(1-x \cos y)$$ is

The order and degree of the differential equation $$\left[1+\left[\frac{d y}{d x}\right]^3\right]^{\frac{7}{3}}=7 \frac{d^2 y}{d x^2}$$ are respective...

The rate at which the metal cools in moving air is proportional to the difference of temperatures between the metal and air. If the air temperature is...

The micro-organisms double themselves in 3 h. Assuming that the quantity increases at a rate proportional to it self, then the number of times it mult...

The particular solution of the differential equation $$y\left(\frac{d x}{d y}\right)=x \log x$$ at $$x=e$$ and $$y=1$$ is

The differential equation obtained from the function $$y=a(x-a)^2$$ is

The differential equation of all lines perpendicular to the line $$5 x+2 y+7=0$$ is

The bacteria increases at the rate proportional to the number of bacteria present. If the original number '$$N$$' doubles in $$4 \mathrm{~h}$$, then t...

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are $$27 \mathrm{~gm}$$ of cer...

The integrating factor of the differential equation $$\left(1+x^2\right) d t=\left(\tan ^{-1} x-t\right) d x$$ is

The particular solution of the differential equation $\log \left(\frac{d y}{d x}\right)=x$, when $x=0, y=1$ is ..............

The solution of the differential equation $y d x-x d y=x y d x$ is ......

The solution of the differential equation $\frac{d \theta}{d t}=-k\left(\theta-\theta_0\right)$ where $k$ is constant, is .............

The order of the differential equation of all circles whose radius is 4 , is ...........

The general solution of $x \frac{d y}{d x}=y-x \tan \left(\frac{y}{x}\right)$ is .............