Mock Test 1 — Differential Equations

Find the order and degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^4 + y^5 = 0$.

Solve the separable first-order differential equation $x(1+y^2)\,dx + y(1+x^2)\,dy = 0$.

The integrating factor for the first-order linear equation $x \frac{dy}{dx} - 2y = x^4$ is:

Find the general solution of the homogeneous equation $\frac{dy}{dx} = \frac{y^2-x^2}{2xy}$.

Solve the first-order linear equation $\frac{dy}{dx} + y = e^{-x}$.

Find the roots of the auxiliary equation for the homogeneous linear system $y'' - 6y' + 8y = 0$.

The general solution of the second-order linear equation $\frac{d^2y}{dx^2} + 4y = 0$ is:

Find the general solution of the Clairaut equation $y = x p + \sqrt{p} \quad \left(p = \frac{dy}{dx}\right)$.

The orthogonal trajectories of the family of concentric circles $x^2 + y^2 = r^2$ are:

If a substance decomposes at a rate proportional to the amount present, and half of it decays in 10 years, the decay constant $k$ matches:

Find the degree of the non-linear derivative radical equation $\left(\frac{d^2y}{dx^2}\right)^2 + \sqrt{\frac{dy}{dx}} + y = 0$.

Find the value of the constant coefficient parameter $b$ if the auxiliary equation for $y'' + b y' + 25y = 0$ has identical repeated roots.

If the integrating factor for $\frac{dy}{dx} + \frac{k}{x}y = \ln x$ is $x^3$, find the integer value of the parameter $k$.

Assertion (A): The degree of the differential equation $\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0$ is strictly equal to 1.
Reason (R): The degree of a differential equation is defined only when the equation can be expressed as a polynomial equation in terms of its derivatives.

Assertion (A): Multiplying the non-exact equation $M \, dx + N \, dy = 0$ by a valid integrating factor $\mu(x,y)$ transforms it into an exact equation that satisfies $\frac{\partial(\mu M)}{\partial y} = \frac{\partial(\mu N)}{\partial x}$.
Reason (R): An integrating factor patches the non-matching partial derivative fields by enforcing exactness across the joint differential layout.