JEE Main · Mathematics

Differential equations

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What are the order and degree (if defined) of the differential equation $(\frac{d ^{2} y}{d x ^{2}})^{3} + (\frac{d y}{d x})^{4} + sin (\frac{d y}{d x}) = 0$ ?
What is the general solution of the differential equation $\frac{d y}{d x} = e^{x - y} + x^{2} e^{- y}$ ?
What is the integrating factor of the first-order linear differential equation $\frac{d y}{d x} + y cot x = 2 x + x^{2} cot x$ ?
If $y (x)$ is the solution of the differential equation $\frac{d y}{d x} + \frac{2 x}{1 + x ^{2}} y = \frac{1}{( 1 + x ^{2} ) ^{2}}$ with initial condition $y (0) = 0$ , what is the value of $y (1)$ ?
The solution of the homogeneous differential equation $\frac{d y}{d x} = \frac{y ^{2} - x ^{2}}{2 x y}$ represents which family of geometric curves?
What is the differential equation representing the family of concentric circles centered at the origin, $x^{2} + y^{2} = r^{2}$ , where $r$ is an arbitrary parameter?
What is the integrating factor of the differential equation $(x + 2 y^{3}) \frac{d y}{d x} = y$ ?
What is the general solution of the differential equation $\frac{d y}{d x} = \frac{x + y + 1}{x + 1}$ for $x > - 1$ ?
What is the general solution of the homogeneous differential equation $x d y - y d x = x^{2} + y^{2} d x$ for $x > 0$ ?
A curve passes through the point $(1, 1)$ and satisfies the property that the slope of the tangent at any point $(x, y)$ is $\frac{2 y}{x}$ . What is the equation of the curve?
What are the order and degree of the differential equation $[1 + (\frac{d y}{d x})^{2}]^{3/2} = k \frac{d ^{2} y}{d x ^{2}}$ , where $k$ is a non-zero constant?
What is the general solution of the differential equation $\frac{d y}{d x} + y = e^{- x}$ ?
If $y (x)$ satisfies the initial value problem $\frac{d y}{d x} + 2 y = 4 x$ with $y (0) = 1$ , what is the value of $y (1)$ ?
What is the differential equation whose general solution is given by $y = A e^{2 x} + B e^{- 2 x}$ , where $A$ and $B$ are arbitrary constants?
What is the general solution of the differential equation $cos x \frac{d y}{d x} + y sin x = 1$ for $x \in (- \frac{π}{2}, \frac{π}{2})$ ?
What is the differential equation of all parabolas having their vertex at the origin and axis along the positive $x$ -axis, $y^{2} = 4 a x$ ( $a > 0$ )?
What is the family of orthogonal trajectories to the family of rectangular hyperbolas $x y = c$ ( $c \neq = 0$ )?
What is the general solution of the differential equation $\frac{d y}{d x} = \frac{x + 2 y}{x}$ for $x > 0$ ?

Differential equations

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