differential equations

Consider the differential equation $\dot{w} = Aw$, with $w(t = 0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$. If $w(t) = e^t\vec{u}_x + e^{-2t}\vec{u}_y$ be the solution to the equati...

The relation between the input current (I) and the output voltage (V) of a circuit is governed by the equation: $C \frac{dV}{dt} = I(t) - m(t)$. The circuit is excited by $I(t) = q...

The function $y(t)$ satisfies $$ t^2 y^{\prime \prime}(t)-2 t y^{\prime}(t)+2 y(t)=0 $$ where $y^{\prime}(t)$ and $y^{\prime \prime}(t)$ denote the first and second derivatives of...

The general form of the complementary function of a differential equation is given by y(t) = (At+B)e^(-2t), where A and B are real constants determined by the initial condition. Th...

The general form of the complementary function of a differential equation is given by $y(t) = (At + B)e^{-2t}$, where $A$ and $B$ are real constants determined by the initial condi...

Consider the differential equation given below. $${{dy} \over {dx}} + {x \over {1 - {x^2}}}y = x\sqrt y $$ The integrating factor of the differential equation is

The general solution of $\frac{d^2 y}{d x^2}-6 \frac{d y}{d x}+9 y=0$ is

Consider the homogeneous ordinary differential equation $x^2 \frac{d^2y}{dx^2} - 3x \frac{dy}{dx} + 3y = 0$, $x > 0$ with $y(x)$ as a general solution. Given that $y(1) = 1$ and $y...

The families of curves represented by the solution of the equation $${{dy} \over {dx}} = - {\left( {{x \over y}} \right)^n}$$ for n = –1 and n = 1 respectively, are

The position of a particle y(t) is described by the differential equation : $${{{d^2}y} \over {d{t^2}}} = - {{dy} \over {dt}} - {{5y} \over 4}$$. The initial conditions are y(0) =...

The general solution of the differential equation $$\,\,{{{d^2}y} \over {d{x^2}}} + 2{{dy} \over {dx}} - 5y = 0\,\,\,$$ in terms of arbitrary constants $${K_1}$$ and $${K_2}$$ is

The ordinary differential equation $$\,\,{{dx} \over {dt}} = - 3x + 2,\,\,$$ with $$x(0)=1$$ is to be solved using the forward Euler method. The largest time step that can be used...

Consider the first order initial value problem $$\,y' = y + 2x - {x^2},\,\,y\left( 0 \right) = 1,\,\left( {0 \le x < \infty } \right)$$ With exact solution $$y\left( x \right)\,\,...

The particular solution of the initial value problem given below is $$\,\,{{{d^2}y} \over {d{x^2}}} + 12{{dy} \over {dx}} + 36y = 0\,\,$$ with $$\,y\left( 0 \right) = 3\,\,$$ and $...

The state variable representation of a system is given as $$$\eqalign{ & \mathop x\limits^ \bullet = \left[ {\matrix{ 0 & 1 \cr 0 & { - 1} \cr } } \right]x;x\left( 0 \right) = \lef...

Consider the differential equation $${{{d^2}x\left( t \right)} \over {d{t^2}}} + 3{{dx\left( t \right)} \over {dt}} + 2x\left( t \right) = 0$$ Given $$x(0) = 20$$ & $$\,x\left( 1 \...

The general solution of the differential equation $$\,\,{{dy} \over {dx}} = {{1 + \cos 2y} \over {1 - \cos 2x}}\,\,$$ is

Which ONE of the following is a linear non - homogeneous differential equation , where $$x$$ and $$y$$ are the independent and dependent variables respectively?

If the characteristic equation of the differential equation $$\,{{{d^2}y} \over {d{x^2}}} + 2\alpha {{dy} \over {dx}} + y = 0\,\,$$ has two equal roots, then the values of $$\alpha...

If $$a$$ and $$b$$ are constants, the most general solution of the differential equation $$\,{{{d^2}x} \over {d{t^2}}} + 2{{dx} \over {dt}} + x = 0$$ is

Match the application to appropriate numerical method Applications $$P1:$$ Numerical integration $$P2:$$ Solution to a transcendental equation $$P3:$$ Solution to a system of linea...

Consider a differential equation $${{dy\left( x \right)} \over {dx}} - y\left( x \right) = x\,\,$$ with initial condition $$y(0)=0.$$ Using Euler's first order method with a step s...

The order of differential equation $$\,\,{{{d^2}y} \over {d{t^2}}} + {\left( {{{dy} \over {dx}}} \right)^3} + {y^4} = {e^{ - t}}\,\,$$ is

Match each differential equation in Group $$I$$ to its family of solution curves from Group $$II.$$ Group $$I$$ $$P:$$$$\,\,\,$$ $${{dy} \over {dx}} = {y \over x}$$ $$Q:$$$$\,\,\,$...

For the differential equation $${{{d^2}y} \over {d{x^2}}} + {k^2}y = 0,$$ the boundary conditions are (i) $$y=0$$ for $$x=0$$ and (ii) $$y=0$$ for $$x=a$$ The form of non-zero solu...

A solution of the differential equation $${{{d^2}y} \over {d{x^2}}} - 5{{dy} \over {dx}} + 6y = 0\,$$ is given by

The following differential equation has $$3{{{d^2}y} \over {d{t^2}}} + 4{\left( {{{dy} \over {dt}}} \right)^3} + {y^2} + 2 = x$$

$$y = {e^{ - 2x}}$$ is a solution of the differential equation $$\,{y^{11}} + {y^1} - 2y = 0$$

Match each of the items A, B and C with an appropriate item from 1, 2, 3, 4 and 5. List - 1 (A) $${a_1}{{{d^{2y}}} \over {d{x^2}}} + {a_2}y{{dy} \over {dx}} + {a_3}y = {a_4}$$ (B)...

Match each of the items $$A, B, C$$ with an appropriate item from $$1, 2, 3, 4$$ and $$5$$ List-$${\rm I}$$ $$(P)$$ $${a_1}{{{d^2}y} \over {d{x^2}}} + {a_2}y{{dy} \over {dx}} + {a_...