List of top Mathematics Questions on Differential equations asked in MHT CET

Solve the differential equation \( \frac{dy}{dx} = \frac{1+y^2}{1+x^2} \) given \(y(0)=1\).

If \[ (2+\sin x)\frac{dy}{dx}+(y+1)\cos x=0 \] and \( y(0)=1 \), then \( y\left(\frac{\pi}{2}\right) \) is equal to:

If \[ \left(\frac{2+\sin x}{1+y}\right)\frac{dy}{dx}=-\cos x \] and \[ y(0)=2, \] then find the value of \[ y\left(\frac{\pi}{2}\right). \]

If \[ (2+\sin x)\frac{dy}{dx}+(y+1)\cos x=0 \] and \[ y(0)=1, \] then find the value of \[ y\left(\frac{\pi}{2}\right). \]

If y = y(x) satisfies the differential equation \[ \left(\frac{2+\sin x}{1+y}\right)\frac{dy}{dx}=-\cos x \] and \( y(0)=2 \), then \( y\left(\frac{\pi}{2}\right) \) is equal to:

The surrounding temperature is \( 20^\circ\text{C} \). A body cools from \( 100^\circ\text{C} \) to \( 60^\circ\text{C} \) in 20 minutes. Find the time taken for the body to cool down to \( 30^\circ\text{C} \).

In a certain culture of bacteria, the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours, then the number of bacteria present in the beginning are:

If $\frac{dy}{dx} = y + 5$ and $y(0) = 4$, then $y(\log 2)$ is equal to

Find the general solution of the differential equation \(\dfrac{dy}{dx} + y\cot x = \csc x\).

Calculate \( y(\log 2) \) if \( \dfrac{dy}{dx} = y + 5 \) and \( y(0) = 4 \).

Find the general solution of the differential equation \( \frac{dy}{dx} + y = e^{-x} \).

Find the general solution of the differential equation \( \dfrac{dy}{dx} + \dfrac{y}{x} = x^2 \).

The differential equation of all circles having their centres on the line \( y = 5 \) and touching the X-axis is

The equation of the curve passing through the origin and satisfying the equation
\[ (1 + x^2) \frac{dy}{dx} + 2xy = 4x^2, \] is

The equation of the curve passing through origin and satisfying $(1 + x^2) \frac{dy}{dx} + 2xy = 4x^2$ is ______.

The general solution of the differential equation $\frac{dy}{dx} = \cot x \cdot \cot y$ is ______.

The differential equation which represents the family of curves $y = c_1 e^{c_2 x}$, where $c_1, c_2$ are arbitrary constants is ______.

The equation of a curve passing through (1,0) and having slope of tangent at any point (x, y) of the curve as $\frac{y-1}{x^2+x}$ is ______.

The solution of the differential equation $(1+x)\frac{dy}{dx}-xy=1-x$ is}

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

The equation of the curve through (0,2) given the sum of ordinate and abscissa at any point exceeds the slope of tangent by 5 is

If \( ^{15}C_4 + ^{15}C_5 + ^{16}C_6 + ^{17}C_7 + ^{18}C_8 = ^{19}C_r \), then the value of \( r \) is equal to}

If \( x = e^{\tan^{-1} \left( \frac{y-x^2}{x^2} \right)} \), then \( \frac{dy}{dx} \) at \( x = 1 \) is

If \( \bar{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}, \bar{b} = \hat{i} - 2\hat{j} - 2\hat{k}, \bar{c} = -\hat{i} + 4\hat{j} + 3\hat{k} \) and if \( \bar{d} \) is vector perpendicular to both \( \bar{b} \) and \( \bar{c} \), \( \bar{a} \cdot \bar{d} = 18 \), then \( |\bar{a} \times \bar{d}|^2 = \)}