General Solution of a Differential Equation

Q.

Find the value of $x^2+y^2$ for which the complex numbers $-3-i{x}^{2}y$ and $x^2+y+4i$ are equal , where x and y are real numbers .

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Q. 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 $x,y\text{and}{y}^{\prime }(\text{here}{y}^{\prime }=\frac{dy}{dx},y^{\prime \prime }=\frac{d^{2}y}{d{x}^2}),$ then which of the following statements is (are) true ?

1. $P=y+x$
2. $P=y-x$
3. $P+Q=1-x+y+y^{\prime }+(y^{\prime }{)}^2$
4. $P-Q=x+y-y^{\prime }-(y^{\prime }{)}^2$

Q. Let the function $y=f\left(x\right)$ satisfy the differential equation $x^2\frac{dy}{dx}=y^2{e}^{1/x}(x\ne 0)$ and $\underset{x\to 0^{-}}{lim}f\left(x\right)=1.$ Identify the CORRECT statement(s) ?

1. Range of $f$ is $(0,1)-\left\{\frac{1}{2}\right\}$
2. $f\left(x\right)$ is bounded
3. $\underset{x\to 0^{+}}{lim}f\left(x\right)=1$
4. $\underset{0}{\overset{e}{\int }}f\left(x\right)dx>\underset{0}{\overset{1}{\int }}f\left(x\right)dx$