GATE EC 2025 Set 1 Q22 — Differential Equations

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

Consider the following partial differential equation $\text{(PDE)}$ $$a \frac{ \partial^{2} f(x ,y)}{\partial x^{2}} + b \frac{ \partial^{2} f(x ,y)}{\partial y^{2}} = f(x, y),$$ where $a$ and $b$ are distinct positive real numbers. Select the combination(s) of values of the real parameters $\xi$ and $\eta$ such that …

Consider the following wave equation, $$\frac{\partial^{2}f(x, t)}{\partial t^{2}} = 10000 \frac{\partial^{2}f(x, t)}{\partial x^{2}}$$ Which of the given options is/are solution(s) to the given wave equation?

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

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

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