Differential Equations

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, then temperature of the body after one hour will be

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

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 radium left after 2 years is

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

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

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

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 mm, then the expression of radius r of the raindrop at any time t is (where 0 \(\le\) t < 3...

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

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

The general solution of the differential equation \(\frac{d y}{d x}=\frac{x+2 y-1}{x+2 y+1}\) 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 differential equation of family of circles whose centres lie on \(\mathrm{X}\)-axis is

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

If \(m\) is order and \(n\) is degree of the differential equation \(\left(\frac{d^2 y}{d x^2}\right)^5+4 \frac{\left(\frac{d^2 y}{d x^2}\right)}{\left(\frac{d^3 y}{d x^3}\right)}+\left(\frac{d^3 y}{d x^3}\right)=x^2\) then

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

The general solution of the differential equation. \(\left(\frac{y}{x}\right) \cos \left(\frac{y}{x}\right) d x-\left[\left(\frac{x}{y}\right) \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)\right] d y=0\) is

The differential equation of the family of circles touching \(y\)-axis at the origin 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 : Differential equations given by I and II are homogeneous differential equations.
S2 : Di...

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 initial quantity of ice. If after 40 minutes the amount of ice left is \(\mathrm{Kx}_0\),...

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

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

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

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 temperature of 32\(^\circ\) F, is given by (where k is a constant of proportionality)

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 lakhs in 30 years, then after another 15 years the population is

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

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

The general solution of the differential equation \(\cos x \cdot \sin y d x+\sin x \cdot \cos y d y=0\) 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 \(p=\frac{d y}{d x}\), then the value of \(m+n\) is

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 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 respectively

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 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 mass of 1.5 gm . Will disintegrate into its mass of 0.5 gm . is proportional to

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

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

The rate of decay of certain substance is directly proportional to the amount present at that instant. Initially, there are \(27 \mathrm{~gm}\) of certain substance and \(3 \mathrm{~h}\) later it is found that \(8 \mathrm{~gm}\) are left, t...

The bacteria increases at the rate proportional to the number of bacteria present. If the original number '\(N\)' doubles in \(4 \mathrm{~h}\), then the number of bacteria in \(12 \mathrm{~h}\) will be

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

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

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 290 K and the metal temperature drops from 370 K to 330 K in 10 min , then the time requi...

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 respectively.

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