PYQs Based Test - 11 : First order LDE (Linear and Nonlinear)

If x(t) and y(t) are the solutions of the system $\frac{dx}{dt}=yand\frac{dy}{dt}=-x$ with the initial conditions x(0) = 1 and y(0) = 1, then x(π/2) + y(π/2) equals _______.

Consider the following differential equation:

$x\left(ydx+xdy\right)\cos \frac{y}{x}=y\left(xdy-ydx\right)\sin \frac{y}{x}$

Which of the following is the solution of the above equation (cis an arbitrary constant)?

Consider the initial value problem below. The value of y at x = In 2, (rounded off to 3 decimal places) is

$\frac{dy}{dx}=2x-y,y\left(0\right)=1$

The families of curves represented by the solution of the equation

$\frac{dy}{dx}=-{\left(\frac{x}{y}\right)}^n$

for n = −1 and n = +1, respectively, are

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