Exams › GATE › Engineering Mathematics

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

1. \(x=t-\tfrac12\)
2. \(x=t^2-\tfrac12\)
3. \(x=\tfrac{t^2}{2}\)
4. \(x=\tfrac{t}{2}\)

Correct answer: \(x=\tfrac{t}{2}\)

Solution

The equation can be written as \(\dfrac{d}{dt}(tx)=t\). Integrating gives \(tx=t^2/2+C\), so \(x=t/2+C/t\). Using \(x(1)=0.5\) gives \(C=0\), hence \(x=t/2\).

Related GATE Engineering Mathematics questions

• The second-order differential equation in the unknown function $u(x,y)$ is defined as $\frac{\partial^2 u}{\partial x^2} = 2$. Assuming $f=f(y)$, $g=g(y)$, and $h=h(y)$, the general solution of the differential equation is
• An ordinary differential equation is given below: $\left(\frac{dy}{dx}\right)(x\ln x)=y$ The solution for the above equation is (where $K$ denotes a constant in the options)
• If k is a constant, the general solution of \(\frac{dy}{dx} - \frac{y}{x} = 1\) will be in the form of
• The integrating factor for the differential equation \(\frac{dP}{dt} + k_2 P = k_1 e^{-k_1 t}\) is
• The respective expressions for the complementary function and particular integral of the differential equation \(\dfrac{d^4y}{dx^4} + 3\dfrac{d^2y}{dx^2} = 108x^2\) are
• The order of the following ordinary differential equation is _____. \[ \frac{d^3 y}{dx^3} + \left(\frac{d^2 y}{dx^2}\right)^6 + \left(\frac{dy}{dx}\right)^4 + y = 0 \]

⚔️ Practice GATE Engineering Mathematics free + battle 1v1 →