Differential Equations

The function y(t) satisfies t²y''(t) - 2ty'(t) + 2y(t) = 0, where y'(t) and y''(t) denote the first and second derivatives of y(t), respectively. Given y'(0) = 1 and y'(1) = −1, the maximum value of y(t) over [0,1] is __...

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 $y(t)$, respectively. Given $y^{\prime}(...

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. The corresponding differential equation is...

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. The corresponding differential equ...

Consider the following partial differential equation (PDE) $$a{{{\partial ^2}f(x,y)} \over {\partial {x^2}}} + b{{{\partial ^2}f(x,y)} \over {\partial {y^2}}} = f(x,y)$$, where a and b are distinct positive real numbers....

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

Which one of the following options contains two solutions of the differential equation $\frac{d y}{d x}=(y-1) x$ ?

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

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

A curve passes through the point ($$x$$ = 1, $$y$$ = 0) and satisfies the differential equation $${{dy} \over {dx}} = {{{x^2} + {y^2}} \over {2y}} + {y \over x}$$. The equation that describes the curve is

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) = 1 and $${\left. {{{dy} \over {dt}}} \rig...

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

Which one of the following is the general solution of the first order differential equation $${{dy} \over {dx}} = {\left( {x + y - 1} \right)^2}$$ , where $$x,$$ $$y$$ are real ?

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 $$\,\,{\left. {{{dy} \over {dx}}} \right|...

The Solution of the differential equation $$\,{{{d^2}y} \over {d{t^2}}} + 2{{dy} \over {dt}} + y = 0\,\,$$ with $$\,y\left( 0 \right) = {y^1}\left( 0 \right) = 1\,\,$$ is

Consider the differential equation $$\,\,{{dx} \over {dt}} = 10 - 0.2\,x$$ with initial condition $$x(0)=1.$$ The response $$x(t)$$ for $$t > 0$$ is

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 $$x(0) = 20$$ & $$\,x\left( 1 \right) = {{10} \over e},$$ where $$e=2.7...

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

The solution of the differential equation $${{h\left( {t + 1} \right)} \over {h\left( t \right)}}\,\,\,\,\,{{{d^2}y} \over {d{t^{ \to 2}}}} + {{2\,dy} \over {dt}} + y\, = \,0$$ with $$\,y\left( 0 \right)\, = \,y'\left( 0...

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

A system is described by the following differential equation, where $$u(t)$$ is the input to the system and $$y(t)$$ is the output of the system. $$$\mathop y\limits^ \bullet \left( t \right) + 5y\left( t \right) = u\lef...

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 $$\alpha $$ are

With initial values $$\,\,\,y\left( 0 \right) = y'\left( 0 \right) = 1,\,\,\,$$ the solution of the differential equation $$\,\,{{{d^2}y} \over {d{x^2}}} - 4{{dy} \over {dx}} + 4y = 0\,\,$$ at $$x=1$$ is ________.

Consider the differential equation $${{{d^2}y\left( t \right)} \over {d{t^2}}} + 2{{dy\left( t \right)} \over {dt}} + y\left( t \right) = \delta \left( t \right)$$ with $$y\left( t \right)\left| {_{t = 0} = - 2} \right.$...

With initial condition $$x\left( 1 \right)\,\,\, = \,\,\,\,0.5,\,\,\,$$ the solution of the differential equation, $$\,\,\,t{{dx} \over {dt}} + x = t\,\,\,$$ is

The solution of differential equation $${{dy} \over {dx}} = ky,y\left( 0 \right) = C$$ is

A function $$n(x)$$ satisfies the differential equation $${{{d^2}n\left( x \right)} \over {d{x^2}}} - {{n\left( x \right)} \over {{L^2}}} = 0$$ where $$L$$ is a constant. The boundary conditions are $$n(0)=k$$ and $$n\le...

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

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

Which of the following is a solution to the differential equation $${d \over {dt}}x\left( t \right) + 3x\left( t \right) = 0,\,\,x\left( 0 \right) = 2?$$

The solution of the differential equation $${k^2}{{{d^2}y} \over {d\,{x^2}}} = y - {y_2}\,\,$$ under the boundary conditions (i) $$y = {y_1}$$ at $$x=0$$ and (ii) $$y = {y_2}$$ at $$x = \propto $$ where $$k$$, $${y_1}$$...

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 solution of $$y$$ (where $$m$$ varies over a...

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

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

Solve the differential equation $${{{d^2}y} \over {d{x^2}}} + y = x\,\,$$ with the following conditions $$(i)$$ at $$x=0, y=1$$ $$(ii)$$ at $$x=0, $$ $${y^1} = 1$$

If $$\,\,\,$$ $$L\left\{ {f\left( t \right)} \right\} = {{s + 2} \over {{s^2} + 1}},\,\,L\left\{ {g\left( t \right)} \right\} = {{{s^2} + 1} \over {\left( {s + 3} \right)\left( {s + 2} \right)}},$$ $$h\left( t \right) =...

$$y = {e^{ - 2x}}$$ 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) $${a_1}{{{d^3}y} \over {d{x^3}}} + {a_2}...

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