The general solution of the differential equation $\frac{dy}{dx} = \frac{2x-3y+4}{3x+2y-7}$ is

Find the family of curves such that the angle between the tangent at any point $(x, y)$ and the tangent to the curve $xy = c^2$ at the point of intersection is $\frac{\pi}{4}$.

The solution of the differential equation $\frac{dy}{dx} = \frac{\sin y + e^x}{\ln y - x \cos y}$ is:

Consider the family of all circles whose centers lie on the straight line $y = x$. If this family of circles is represented by the differential equation $P y^{\prime \prime} + Q y^{\prime} + 1 = 0$,where $P, Q$ are functions of $x, y$ and $y^{\prime}$ (here $y^{\prime} = \frac{dy}{dx}, y^{\prime \prime} = \frac{d^2y}{dx^2}$),then which of the following statements is (are) true?
$(A) P = y + x$
$(B) P = y - x$
$(C) P + Q = 1 - x + y + y^{\prime} + (y^{\prime})^2$
$(D) P - Q = x + y - y^{\prime} - (y^{\prime})^2$

The solution of $\frac{d^2y}{dx^2} = \sec^2 x + x e^x$ is

The solution of the differential equation $\frac{d^2 y}{d x^2}+y=0$ is