Mock Test 2 — Differential Equations

Solve the non-linear first-order equation $\frac{dy}{dx} = \frac{x+y+1}{2x+2y+1}$ using a targeted linear combination substitution.

Find the general solution of the Bernoulli differential equation $\frac{dy}{dx} + \frac{1}{x}y = x^2 y^2$.

Find the integrating factor $\mu(x)$ for the non-exact equation $(x^2+y^2+x)\,dx + xy\,dy = 0$ if it depends solely on $x$.

Find the singular solution to the Clairaut equation $y = x p + \frac{1}{4p^2} \quad \left(p = \frac{dy}{dx}\right)$.

Find the particular integral $y_p$ for the second-order linear non-homogeneous equation $y'' - 3y' + 2y = e^{3x}$.

The solution curve to the equation $\frac{dy}{dx} = \frac{y \ln y}{x}$ passing through the coordinate initial point $(1, e)$ matches:

Find the orthogonal trajectories of the family of parabolas $y^2 = 4ax$.

A mixing tank contains $200\text{ L}$ of fluid holding $20\text{ kg}$ of dissolved chemical salt. Brine containing $0.05\text{ kg}$ of salt per liter pumps in at $4\text{ L/min}$, and the mixed solution drains out at the same rate. Find the limiting salt content in the tank as $t \to \infty$.

Find the complete general solution of the second-order linear differential equation $y'' + 2y' + 5y = 0$.

If the forcing function for the second-order system $y'' - 4y' + 4y = e^{2x}$ matches a repeated root of the auxiliary equation, the particular integral template $y_p$ must be modeled as:

Find the total number of arbitrary constants that must appear in the general solution of a fourth-order ordinary differential equation.

If the orthogonal trajectories of the family of curves $y^2 = c x^3$ are ellipses modeled by the equation $2x^2 + n y^2 = a^2$, find the integer value of the scaling parameter $n$.

Find the value of the root parameter $m$ if $y = e^{mx}$ is a solution to the homogeneous equation $y'' - 7y' + 12y = 0$ and $m$ is the smaller root.

Assertion (A): The singular solution of a Clairaut differential equation cannot be obtained by substituting any real value for the arbitrary constant inside its general solution equation.
Reason (R): A singular solution represents the tangent envelope profile bounding the general linear family curves, meaning it satisfies the differential equation without belonging to its parametric curve family.

Assertion (A): The variation of parameters method is a general technique that can find the particular integral for any second-order linear non-homogeneous differential equation, regardless of whether the forcing function is transcendental or rational.
Reason (R): This method does not rely on guessing a solution template; instead, it uses the components of the complementary function to construct a general integral formula.