initial value problem

Consider the following differential equation $\frac{\partial y}{\partial x} = 3\frac{\partial y}{\partial t} + y$ If $y(x, 0) = 10e^{-2x}$, then the solution of the differential eq...

Let y be the solution of the differential equation with the initial conditions given below. If $y(x = 2) = A \ln 2$, then the value of A is ________ (rounded off to 2 decimal place...

Let $y$ be the solution of the differential equation with the initial conditions given below. If $y(x=2)=A \ln 2$, then the value of $A$ is _________ (rounded off to 2 decimal plac...

If $x(t)$ satisfies the differential equation $t\frac{dx}{dt} + (t - x) = 0$ subject to the condition $x(1) = 0$, then the value of $x(2)$ is ________ (rounded off to 2 decimal pla...

If $x(t)$ satisfies the differential equation $t \frac{dx}{dt} + (t - x) = 0$ subject to the condition $x(1) = 0$, then the value of $x(2)$ is __________ (rounded off to 2 decimal...

Consider the second-order linear ordinary differential equation $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} - y = 0, x \geq 1$ with the initial conditions $y(x = 1) = 6, \frac{dy}{dx}...

Consider the second-order linear ordinary differential equation $\rm x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-y=0, x\ge1$ with the initial conditions $\rm y(x=1)=6, \left.\frac{dy}{dx}\...

If y(x) satisfies the differential equation (sin x) dy/dx + y cos x = 1, subject to the condition y(π/2) = π/2, then y(π/6) is

Consider the following differential equation $(1 + y) \frac{dy}{dx} = y$. The solution of the equation that satisfies the condition $y(1) = 1$ is

For the equation $\frac{dy}{dx} + 7x^2y = 0$, if $y(0) = 3/7$, then the value of $y(1)$ is

If $y$ is the solution of the differential equation $y^3 \frac{dy}{dx} + x^3 = 0$, $y(0)=1$, the value of $y(-1)$ is

Consider the differential equation $3y''(x)+27y(x)=0$ with initial conditions $y(0)=0$ and $y'(0)=2000$. The value of y at x = 1 is _________.

Given the ordinary differential equation $\frac{d^2y}{dx^2} + \frac{dy}{dx} -6y=0$ with y(0) = 0 and $\frac{dy}{dx}(0) = 1$, the value of y(1) is ________ (correct to two decimal p...

Consider the differential equation $$\,\,3y''\left( x \right) + 27y\left( x \right) = 0\,\,$$ with initial conditions $$y\left( 0 \right) = 0$$ and $$y'\left( 0 \right) = 2000.\,\,...

Consider the following differential equation $${{dy} \over {dt}} = - 5y;$$ initial condition: $$y=2$$ at $$t=0.$$ The value of $$y$$ at $$t=3$$ is

The solution of the initial value problem $$\,\,{{dy} \over {dx}} = - 2xy;y\left( 0 \right) = 2\,\,\,$$ is

The function $$f(t)$$ satisfies the differential equation $${{{d^2}f} \over {d{t^2}}} + f = 0$$ and the auxiliary conditions, $$f\left( 0 \right) = 0,\,{{df} \over {dt}}\left( 0 \r...

The solution of $$x{{dy} \over {dx}} + y = {x^4}$$ with condition $$y\left( 1 \right) = {6 \over 5}$$

Given that $$\mathop x\limits^{ \bullet \bullet } + 3x = 0$$ and $$x\left( 0 \right) = 1,\,\,\mathop x\limits^ \bullet \left( 0 \right) = 1,$$ What is $$x(1)$$ ________.

The solution of $${{d\,y} \over {d\,x}} = {y^2}$$ with initial value $$y(0)=1$$ is bounded in the interval is

The solution of the differential equation $${{dy} \over {dx}} + 2xy = {e^{ - {x^2}}}\,\,$$ with $$y(0)=1$$ is

If $${x^2}\left( {{{d\,y} \over {d\,x}}} \right) + 2xy = {{2\ln x} \over x}$$ and $$y(1)=0$$ then what is $$y(e)$$?

Find the solution of the differential equation $$\,{{{d^2}u} \over {d{t^2}}} + {\lambda ^2}y = \cos \left( {wt + k} \right)$$ with initial conditions $$\,y\left( 0 \right) = 0,\,\,...

Solve the initial value problem $${{{d^2}y} \over {d{x^2}}} - 4{{dy} \over {dx}} + 3y = 0$$ with $$y=3$$ and $${{dy} \over {dx}} = 7$$ at $$x=0$$ using the laplace transform techni...

For the differential equation $$\,\,{{dy} \over {dt}} + 5y = 0\,\,$$ with $$y(0)=1,$$ the general solution is