List of top Engineering Mathematics Questions on Differential Equations

The following differential equation governs the evolution of variable \(x(t)\) with time \(t, t \geq 0\).
\(\frac {d^2x}{dt^2} + 4x = e^{-t}\)
Given the initial conditions \(x = 0\) and \(\frac {dx}{dt} = 0\) at \(t = 0\), the value of \(x\) at \(t =\frac {\pi}{8}\) is _____  . (Rounded off to 3 decimal places)

Consider the following function:
\[ f(x) = \begin{cases} k, & \text{if } x = 1 \\\frac{\sqrt{3x + 1} - \sqrt{2x + 2}}{x - 1}, & \text{if } x > -\frac{1}{3}, x \neq 1 \end{cases} \]
If \( f(x) \) is continuous at \( x = 1 \), the value of \( k \) (correct up to 2 decimal places) is

A scientist wants to find the root of the equation \(2x^3 + x^2 - 1 = 0\) lying in \((0,1)\). He applies the Secant method only once by taking two initial guesses, 0.5 and 0.7. The value of the root is approximately:

The differential equation \( \frac{dy}{dx} = \frac{M(x,y)}{N(x,y)} \) is exact if _______. (

Consider a system of the following two partial differential equations :
\(\frac{∂\alpha}{∂x}=-2\frac{∂\beta}{∂t}\)
\(\frac{∂\beta}{∂x}=-2\frac{∂\alpha}{∂t}\)
Which one of the following choices is a possible solution for the system ?

To solve the equation x = 2 cos x using Newton-Raphson's method, which one of the following iterations should be used ?

If the complete solution of the ordinary differential equation
\(\frac{d^2y}{dx^2}+a\frac{dy}{dx}+\beta y=0\;is\;y=c_1e^{-x}+c_2e^{-3x},\)
Then the values of \(\propto\) and B are: