Differential Equations - Class 12

Let $z_1, z_2, \dots, z_6$ be the distinct roots of the equation $z^6 - 7iz^3 - 12 = 0$. Consider the differential equation $\left( \sum_{k=1}^{6} |z_k|^2 \right) \det(M) + e^{\frac{d^2y}{dx^2}} + x \frac{dy}{dx} = 0$ where $M$ is the matrix $M = \begin{pmatrix} 0 & i \left(\frac{d^4y}{dx^4}\right)^3 & \left(\frac{dy}{dx}\right)^2 \\ -i \left(\frac{d^4y}{dx^4}\right)^3 & 0 & x^3 \\ -\left(\frac{dy}{dx}\right)^2 & -x^3 & 0 \end{pmatrix}$ Which of the following represents the order and degree of the given differential equation?

Let $y = y(x)$ be a sufficiently differentiable function satisfying the equation $\lim_{t \to x} \frac{1}{\ln(1 + t - x)} \int_{x}^{t} \left( \frac{d^2y}{du^2} \right)^{3/5} du = \int_{0}^{x} y(v) \sqrt{v} \, dv$ for $x > 0$. If the lowest-order pure differential equation (free from integrals) satisfied by $y(x)$ has order $O$ and degree $D$, then the value of $O + D$ is:

Let $y = y(x)$ be a function satisfying the differential equation: $\frac{ \left( \frac{d^2y}{dx^2} \right)^2 - \left( \frac{dy}{dx} \right)^2 }{ \frac{d^2y}{dx^2} - \frac{dy}{dx} } = \frac{d^2y}{dx^2} + \frac{dy}{dx} + \left( \frac{dy}{dx} \right)^3 - x$ If $O$ and $D$ represent the order and degree of this differential equation respectively, then the ordered pair $(O, D)$ is:

Let $T_n, U_n,$ and $V_n$ be the $n^{\text{th}}$ terms of three sequences whose sums of the first $n$ terms are given by $S_n = \frac{n}{3}(n^2 - 9n + 26)$, $S'_n = \frac{n}{6}(2n^2 - 15n + 37)$, and $S''_n = \frac{n}{6}(2n^2 - 9n + 13)$ respectively. Consider the differential equation: $T_n \left( \frac{d^3 y}{dx^3} \right)^2 + U_n \left( \frac{d^2 y}{dx^2} \right)^4 + V_n \left( \frac{dy}{dx} \right)^5 + y = 0$ Let $O_n$ and $D_n$ denote the order and degree of this differential equation for a given $n \in \mathbb{N}$. Find the value of $\sum_{n=1}^{5} (O_n \times D_n)$.

Let $P$ be a $3 \times 3$ orthogonal matrix and $N$ be a $3 \times 3$ nilpotent matrix of index 3. Consider the differential equation given by $\det \left( P^T \left[ \left( \frac{d^3y}{dx^3} \right)^{\frac{2}{3}} I + \left( \frac{d^2y}{dx^2} \right)^{\frac{3}{4}} N + \left( \frac{dy}{dx} \right)^{\frac{4}{5}} N^2 \right] P \right) = x^2 + 1$ where $I$ is the $3 \times 3$ identity matrix. If $m$ and $n$ are the order and degree of this differential equation respectively, find the value of $m + n$.

Let $P(r) = r^4 - 8r^2 + 16$. Let $r_1, r_2, r_3, r_4$ be the roots of the equation $P(r) = 0$. Let $V_1$ be the vector space over $\mathbb{R}$ spanned by the set of functions $S_1 = \{e^{r_1 x}, e^{r_2 x}, e^{r_3 x}, e^{r_4 x}\}$. Let $V_2$ be the solution space of the differential equation $\frac{d^4y}{dx^4} - 8\frac{d^2y}{dx^2} + 16y = 0$. Let $d_1 = \dim(V_1)$ and $d_2 = \dim(V_2)$. Let $O$ be the order of the lowest order linear homogeneous differential equation with constant coefficients whose solution space is exactly $V_1$. Let $W(x)$ be the Wronskian of the fundamental solutions $e^{2x}, x e^{2x}, e^{-2x}, x e^{-2x}$ of the above differential equation. Find the value of $d_1 \times d_2 \times O + W(0)$.

Let $\Gamma$ be the family of curves satisfying the differential equation $2xy \frac{dy}{dx} = y^2 - x^2 + c^2$, where $c > 0$ is a fixed parameter. A new family of curves $\Gamma^*$ is obtained by reflecting each curve in $\Gamma$ across the line $y = c$. Let $C \in \Gamma^*$ be a curve that is tangent to the $x$-axis. Find the area of the region enclosed by $C$.

A sequence of functions $\{y_n(x)\}_{n=1}^\infty$ satisfies the differential equation $\frac{dy_n}{dx} - \lambda y_n = 0$ with the initial condition $y_n(0) = C_n(\lambda)$, where $\lambda > 0$ is a parameter. It is given that $C_0(\lambda) = 0$ and for all $n \ge 1$, the constants $C_n(\lambda)$ satisfy the relation: $\int_0^1 y_n(x) dx - \int_0^1 y_{n-1}(x) dx = \frac{e^\lambda + \lambda - 1}{\lambda^4 n(n+1)}$ Let $C_\infty(\lambda) = \lim_{n \to \infty} C_n(\lambda)$. Find the value of $\lim_{\lambda \to 0^+} (e^\lambda - 1) C_\infty(\lambda)$.

Let $\vec{u}, \vec{v}, \vec{w}$ be a right-handed system of mutually orthogonal unit vectors in $\mathbb{R}^3$. A family of curves $C_1$ in the plane spanned by $\vec{u}$ and $\vec{v}$ is given by the position vector $\vec{r} = x\vec{u} + y\vec{v}$ satisfying the differential equation: $d\vec{r} = \left( 3x^2 [\vec{u}\ \vec{r}\ \vec{w}] \right) \vec{u} + \left( 2y [\vec{r}\ \vec{u}\ \vec{w}] \right) \vec{v}$ where $[\vec{a}\ \vec{b}\ \vec{c}]$ denotes the scalar triple product. Let $C_2$ be the orthogonal trajectory of the family $C_1$ that passes through the point with position vector $\vec{u} + 5\vec{v}$. If the point $2\vec{u} + \alpha\vec{v}$ lies on $C_2$, find the value of $\alpha^2$.

Let $f: (0, 6) \setminus \{3\} \to \mathbb{R}$ be a differentiable function satisfying the differential equation $(x-3) \frac{dy}{dx} + (x^2 - 6x + 9) y = 0$ with the initial condition $f(2) = e^4$. Let $A$ be the number of points of inflection of $f(x)$ in its domain. Let $B = \lim_{x \to 3} \ln(f(x))$. Let $C = 1$ if $f(x)$ attains a global maximum on the interval $[1, 5] \setminus \{3\}$, and $C = 0$ otherwise. Find the value of $2A + 2B + C$.

Let $y = y(x)$ be a continuously differentiable function on $(0, 2\pi)$ satisfying the differential equation $\sin x \frac{dy}{dx} - y \cos x = \sin x \sqrt{1 - \cos^2 x}$ with the condition $y\left(\frac{\pi}{2}\right) = \frac{\pi}{2}$. Then the value of $y\left(\frac{3\pi}{2}\right)$ is:

Let $f: \mathbb{R} \to (0, \infty)$ be a differentiable function satisfying $f(0) = 1$ and $f'(x) = f(x) \left( 1 + \frac{2x}{x^2+1} \right) + \frac{(x^2+1)^2}{f(x)} - 2(x^2+1)$ for all $x \in \mathbb{R}$. Find the value of $f(3)$.

Let $y = y(x)$ be the solution to the differential equation $x \frac{dy}{dx} = y + x + 3|x - y|$ for $x > 0$, satisfying the initial condition $y(1) = 0$. Find the value of $y(2)$.

Let a curve $C$ satisfy the differential equation $(2x - y - 1)dx + (2y - x - 1)dy = 0$. If the curve passes through the point $(3, 2)$, then the area of the region enclosed by $C$ is

Let $y = y(x)$ be the solution to the differential equation $x dy - y dx = \sqrt{x^2+y^2} \left[ \lim_{t \to 0} \frac{\sin\left( t \left( \tan^{-1}\left(\frac{x}{y}\right) - \cot^{-1}\left(\frac{y}{x}\right) \right) \right)}{t} \right] dx$ for $x > 0$, satisfying the initial condition $y(1) = -1$. Then the value of $y\left( (\sqrt{2}+1)^{1/\pi} \right)$ is

Let $y = y(x)$ be the solution to the differential equation $x(y-x)\frac{dy}{dx} = y^2 - xy - x^2$ for $x > 0$, satisfying the initial condition $y(1) = 0$. Find the value of $y(e^{3/8})$.

A moving point $P(x,y)$ in the first quadrant traces a curve $C$ passing through the point $\left(\frac{1}{2}, \frac{\sqrt{7}}{2}\right)$. The slope of the tangent to $C$ at any point $P$ is given by the differential equation $\frac{dy}{dx} = \frac{y^2 - x^2}{2xy}$. Furthermore, the locus of $P$ is constrained such that its distance from the origin is strictly less than its distance from the line $x = 3$. Let $S$ be the set of all possible $y$-coordinates of the point $P$. If the supremum of $S$ is $M$, what is the value of $M^2$?

Let $y = y(x)$ be the solution to the differential equation $x \frac{dy}{dx} = y + x \sqrt{1 - \sin\left(\frac{y}{x}\right)}, \quad x > 0$ with the initial condition $y(1) = \pi$. If $y(x_0) = \frac{3\pi}{2} x_0$ for some $x_0 > 1$, then the value of $x_0$ is:

Let $A(x, 0)$ (where $x > 0$) be a point and $L$ be its polar with respect to the circle $x^2 + y^2 = 1$. Let $B$ be the intersection of $L$ with the $x$-axis. A function $P(x)$ is defined as $P(x) = \left| 1 - \frac{OB}{OA} \right|$, where $O$ is the origin. Let $y = y(x)$ be the solution to the differential equation $\frac{dy}{dx} + P(x) y = 1 - \frac{1}{x^2}$ for $x > 0$, satisfying $y\left(\frac{1}{2}\right) = e^{5/2} - 1$. Find the value of $y(2)$.

Let $y=y(x)$ be the solution to the differential equation $\cos x \ln(\sec x + \tan x) \frac{dy}{dx} + y = \tan x \ln(\sec x + \tan x)$ for $x \in \left(0, \frac{\pi}{2}\right)$, satisfying the initial condition $y\left(\frac{\pi}{4}\right) = \sqrt{2} - \frac{1}{\ln(\sqrt{2} + 1)}$. Then $y\left(\frac{\pi}{3}\right)$ is equal to:

Let a curve be defined parametrically by $x(t) = t^2 + t$ and $y = y(t)$ for $t \ge 0$. It is given that $y(x)$ satisfies the differential equation $(4x+1)\frac{dy}{dx} - 2y = \frac{1}{2}(4x+1)^{3/2}$ with $y(0) = \frac{1}{2}$. According to Lagrange's Mean Value Theorem, there exists a point on the curve for $x \in [0, 2]$ where the tangent is parallel to the chord joining the endpoints of the curve on this interval. The value of the parameter $t$ corresponding to this point is:

Let $Q(x) = \sum_{k=0}^{3} \binom{3}{k} (-1)^k x^{2k} (\text{sgn}(x))^k$, where $\text{sgn}(x)$ denotes the signum function. Let $y = y(x)$ be a differentiable function satisfying the differential equation $(1 - x|x|) \frac{dy}{dx} + 6 |x| y = x (Q(x))^3$ for all $x \in \mathbb{R}$, with the initial condition $y(0) = 0$. Find the value of $y(-1)$.

Let $y = y(x)$ be the continuous solution to the differential equation $\frac{dy}{dx} + x|x| y = x^2 e^{x^3/3}$ for $x \in \mathbb{R}$, satisfying $y(\sqrt[3]{3}) = \frac{e^2 + 1}{2e}$. Then the value of $y(-\sqrt[3]{6})$ is

Let $y = f(x)$ be the solution to the initial value problem $\frac{dy}{dx} + \frac{1}{x \ln x} y = \frac{1}{x^2 |\ln x|}$ with $f(e) = 1 - \frac{1}{e}$, defined on its maximal interval of existence. Let $S$ be the set of all possible values of $f(1/e)$. Find $S$.

Let $y = y(x)$ be the solution to the differential equation $x \frac{dy}{dx} + (1 - 2x|x-1|) y = x^2(1-x) y^2$ for $x > 0$, satisfying the condition $\lim_{x \to 0} x y(x) = \frac{2}{2e - 1}$. Find the value of $y(2)$.

Let $y = f(x)$ be a curve in the $xy$-plane satisfying the differential equation $\frac{dy}{dx} - y = x$. Let $P(x, f(x), 0)$ be a point on the curve and $d(x)$ be its perpendicular distance to the plane $\Pi: x + y + \sqrt{2}z - 3 = 0$. Given that $d(0) = 3$ and the scalar projection of the position vector $\vec{OP}$ at $x=0$ (where $O$ is the origin) onto the vector $\vec{n} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}$ is negative, find $f(\ln 2)$.

Let $A = \begin{pmatrix} -3 & -4 & 4 \\ -3 & -1 & 3 \\ -9 & -8 & 10 \end{pmatrix}$ and $X(t)$ be the solution to the system of differential equations $\frac{dX}{dt} = A X$ with initial condition $X(0) = \begin{pmatrix} 2 \\ 3 \\ 6 \end{pmatrix}$. A vector $Y(t)$ is defined implicitly by the relation $M(t) Y(t) = X(t)$, where $M(t) = A^2 - t A + (t^2 - 7)I$. Let $S$ be the set of all distinct times $t > 0$ for which the explicit conversion to $Y(t)$ fails (i.e., $M(t)$ is singular). If $\alpha = \sum_{t \in S} t$, find the value of $\frac{1}{\alpha^2} \sqrt{\text{Tr}(X(\ln \alpha) X^T(\ln \alpha))}$.

Let $C_1$ be a curve defined by the differential equation $(x^2+4)dy + (2xy - 24)dx = 0$, passing through the point $(0, 12)$. Let $P$ be the point of inflection of $C_1$ having the largest $x$-coordinate. A hyperbola $H$, with its transverse axis along the $x$-axis, has an asymptote in the first quadrant whose slope is equal to the absolute value of the slope of the tangent to $C_1$ at $P$. What is the eccentricity of the hyperbola $H$?

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable function such that $f(2) = 2$, $f(3) = 2$, and for all $x \in \mathbb{R}$, $(f(x) - |x-2| - |x-4|) \left( f'(x) + 6x^2 - 36x + 48 \right) = 0$ Find the value of $f(1) + f(5)$.

Let $X(t)$ be a $3 \times 3$ matrix of differentiable functions satisfying the differential equation $X'(t) = A(t)X(t) + X(t)B(t)$ for $t \in (0, \pi)$, with the initial condition $X\left(\frac{\pi}{2}\right) = I_3$. Given that: $A(t) = \begin{pmatrix} t & \sin t & \cos t \\ -\sin t & -t & e^t \\ -\cos t & -e^t & \frac{2t}{t^2+1} \end{pmatrix}, \quad B(t) = \begin{pmatrix} \cot t & t^2 & \ln t \\ -t^2 & \cot t & \sqrt{t} \\ -\ln t & -\sqrt{t} & \cot t \end{pmatrix}$ If $f(t) = \det(X(t))$, find the value of $\lim_{t \to 0^+} \frac{f(t)}{t^3}$.

Let $\vec{a}$ and $\vec{b}$ be two linearly independent vectors in $\mathbb{R}^3$ such that $|\vec{a}| = 2$ and the scalar projection of $\vec{b}$ on $\vec{a}$ is $-\frac{3}{2}$. A differentiable vector-valued function $\vec{y}(x)$ satisfies the differential equation $(x + \vec{a} \cdot \vec{b}) \frac{d\vec{y}}{dx} = \vec{a} + (x + \vec{a} \cdot \vec{b}) \vec{b}$ for $x < 3$, with the initial condition $\vec{y}(1) = \vec{b}$. Find the value of $\vec{y}(2) \cdot \vec{a}$.

Let $f(x)$ be a continuous function on $[0, 3]$ such that $f(0) = 0$ and $f'(x) = \frac{|x-1|}{x-1} \frac{2x}{x^2+1}$ for all $x \in (0, 3) \setminus \{1\}$. Let $S$ be the set of all $c \in (0, 3)$ that satisfy the equation $f'(c) = \frac{f(3) - f(0)}{3}$. Find the number of elements in $S$.

Let $\alpha = \int_{-1}^{1} x^2 \sin(x^3) \,dx$. Consider the differential equation for $x > 0, y > 0$: $x \frac{dy}{dx} - y = x \left( \left[ \frac{y}{x} \right] - 1 + \alpha \right) \left( \frac{y}{x} - 2 \right)$ where $[ \cdot ]$ denotes the greatest integer function. Let $y_c(x)$ denote the solution curve satisfying the initial condition $y(1) = c$. Which of the following expressions for $y_c(x)$ (valid in a neighborhood of $x=1$) is INCORRECT?

Let $C_t$ be a family of curves satisfying the differential equation $\frac{dy}{dx} = (x - t)e^x$, where $t \in \mathbb{R}$ is a parameter. For each $t$, the curve $C_t$ passes through the origin. Let $P_t$ be the point of local extremum of the curve $C_t$. As $t$ varies, $P_t$ traces a curve $L_1$. A new curve $L_2$ is defined as the locus of a point $Q$, which is obtained by applying the transformation matrix $M = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$ to the coordinates of $P_t$ (treating $P_t$ as a column vector). Let $A$ be the area of the region bounded by $L_2$, the normal to $L_2$ at its point of local maximum, and the coordinate axes. Find the value of $A$.

Let $M = \begin{pmatrix} 2 & -1 & 1 \\ 1 & 2 & -1 \\ 3 & 1 & 2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 3 \\ 2 \\ 11 \end{pmatrix}$. Let $\begin{pmatrix} x_0 \\ y_0 \\ z_0 \end{pmatrix}$ be the unique solution to the system $M \vec{X} = \vec{b}$. Consider the system of differential equations $\frac{d\vec{v}}{dt} = A\vec{v}$, where $A = \begin{pmatrix} -1 & -3 \\ 3 & -1 \end{pmatrix}$ and $\vec{v}(t) = \begin{pmatrix} x(t) \\ y(t) \end{pmatrix}$. Given the initial condition $\vec{v}(0) = \begin{pmatrix} x_0 \\ y_0 \end{pmatrix}$, let $L$ be the total arc length of the curve traced by $\vec{v}(t)$ in the $xy$-plane as $t$ varies from $0$ to $\infty$. Find the value of $L^2$.

Let $S$ be the set of all continuously differentiable functions $f: [0, \pi] \to \mathbb{R}$ satisfying $(f'(x))^2 = \sin^2(2x)$ and $f(0)=0$. Let $R$ be the region bounded by all the curves in $S$. Let $S'$ be the subset of $S$ consisting of functions that do NOT satisfy the condition $\int_0^{\pi/2} f(x) \cos(2x) \, dx \ge 0$, and let $R'$ be the region bounded by the curves in $S'$. If $A$ and $A'$ are the areas of regions $R$ and $R'$ respectively, what is the value of $A - A'$?

Let $y(x)$ be a real-valued function satisfying the differential equation $x \frac{dy}{dx} = y + \frac{x^2}{y} f(x)$ for $x \in \mathbb{R} \setminus \{0\}$, where $f(x)$ is a continuous odd function such that $\lim_{x \to 0} \frac{f(x)}{x}$ exists. The solution is given by $y^2 = x^2 \left( 2 \int_0^x \frac{f(t)}{t} dt + C(x) \right)$, where the integration parameter $C(x)$ is defined as $C(x) = \lambda [x]^2 + \mu [x]$, with $[ \cdot ]$ denoting the greatest integer function, and $\lambda, \mu \in \mathbb{R}$. Let $I_n = \int_{-n}^{n} \frac{y^2}{x^2} dx$. If $I_2 = 2$ and $I_3 = 13$, find the value of $\int_0^{\lambda + \mu} (3t^2 - 2t) dt$.

Let $y = y(x)$ be a thrice differentiable function on $(0, \pi)$ satisfying the differential equation: $\begin{vmatrix} y & \sin x & \cos x \\ y''' & -\cos x & \sin x \\ y' & \cos x & -\sin x \end{vmatrix} \sin x + \begin{vmatrix} y & \sin x & \cos x \\ y' & \cos x & -\sin x \\ y'' & -\sin x & -\cos x \end{vmatrix} \cos x = 0$ Given that $y\left(\frac{\pi}{2}\right) = 1$, $y'\left(\frac{\pi}{2}\right) = 1$, and $y''\left(\frac{\pi}{2}\right) = 0$, find the value of $y\left(\frac{\pi}{6}\right)$.

A curve $y = f(x)$ passes through the point $(1, 0)$. For all $x \ge 1$, the slope of the normal to the curve at any point $(x, y)$ is given by $-\left( \int_0^x [t] dt \right)^{-1}$, where $[ \cdot ]$ denotes the greatest integer function. Let $L$ be the tangent to the curve at $x = \frac{5}{2}$. If $L$ intersects the x-axis at $A$ and the y-axis at $B$, find the length of the line segment $AB$.

Let $y(x)$ be the positive solution to the differential equation $\frac{dy}{dx} + \frac{1}{2} y = \frac{1}{2} (x^5 - 15x^4 + 75x^3 - 160x^2 + 150x - 20) y^3$ for $x > 0$, with the initial condition $y(0) = \frac{1}{2\sqrt{5}}$. Let $S$ be the set of all points $x \in (0, \infty)$ where $y(x)$ attains a local extremum. Let $k = \sum_{x \in S} x$. Consider the polynomial $f(x) = x^3 - k x^2 + 11x - 6$. Which of the following statements is true?

Let $A$ be a $3 \times 3$ symmetric orthogonal matrix. Let $\vec{y}(x)$ be a differentiable vector-valued function in $\mathbb{R}^3$ satisfying the differential equation: $x A \frac{d\vec{y}}{dx} + A \vec{y} = x^2 (\vec{y}^T A \vec{y}) A \vec{y}, \quad x > 0$ If $\vec{y}(1)^T A \vec{y}(1) = \frac{1}{c}$ for some constant $c$, which of the following represents $w(x) = \vec{y}^T A \vec{y}$?

Find the equation of the orthogonal trajectories of the family of curves $x = k |y|^a e^{b|y|}$, where $k$ is an arbitrary non-zero constant, and $a, b$ are positive constants.

Consider the family of curves given by the differential equation $\frac{dy}{dx} = \frac{a^2+\sin^2 x}{y - y^3}$, where $a > 0$. Let $y=f(x)$ be the orthogonal trajectory to this family. If $f(0) = y_0$ (where $y_0 \notin \{-1, 0, 1\}$), find the value of $\lim_{x \to \pi} \frac{|1 - (f(x))^{-2}|}{|1 - y_0^{-2}|}$.

Let $y = f(x)$ be the particular solution to the differential equation $\frac{dy}{dx} = y^2 (x^2 - 1)$ satisfying the initial condition $f(-1) = -1$. Let $I$ be the maximal interval of existence of $f(x)$ containing $x = -1$. Which of the following represents the range of $f(x)$ on the interval $I$?

Let $A = \frac{1}{3} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & b & c \end{pmatrix}$ be an orthogonal matrix with $\det(A) = 1$. Let $B = 3A - 3I$, where $I$ is the $3 \times 3$ identity matrix. A differentiable vector function $\mathbf{x}(t) = \begin{pmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{pmatrix}$ satisfies the differential equation $\frac{d\mathbf{x}}{dt} = B \mathbf{x}(t) + \begin{pmatrix} 1 \\ -1 \\ 2 \end{pmatrix}$ with the initial condition $\mathbf{x}(0) = \begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix}$. Find the value of $\lim_{t \to \infty} \left( 3x_1(t) + 6x_2(t) + 12x_3(t) \right)$.