Integrating Factor

Q.

Which of the following is true for $\mathrm{y}\left(\mathrm{x}\right)$ that satisfies the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{xy}-1+\mathrm{x}-\mathrm{y};\mathrm{y}\left(0\right)=0$?

1. $\mathrm{y}\left(1\right)=1$

2. $\mathrm{y}\left(1\right)={\mathrm{e}}^{\frac{1}{2}}-1$

3. $\mathrm{y}\left(1\right)={\mathrm{e}}^{\frac{1}{2}}-{\mathrm{e}}^{-\frac{1}{2}}$

4. $\mathrm{y}\left(1\right)={\mathrm{e}}^{-\frac{1}{2}}-1$

Q.

A solution of the differential equation ${\left(\frac{dy}{dx}\right)}^2-x\left(\frac{dy}{dx}\right)+y=0$ is

1. $y=2$

2. $y=2x$

3. $y=2x-4$

4. $y=2x^2-4$

Q. The solution of the differential equation $(2x-y+2)dx+(4x-2y-1)dy=0$ is
(where $k$ is integration constant)

1. $|2x+y|=k{e}^{-(2x-y)}$
2. $|2x-y|=k{e}^{-(x+2y)}$
3. $|x+2y|=k{e}^{(2x-y)}$
4. $(2x-y)=k{e}^{|x-2y|}$

Q.

An integrating factor for the differential equation
$1+x^2)dx-(ta{n}^{-1}y-x)dy=0$

[MP PET 1993]

1. $ta{n}^{-1}y$

2. $e^{ta{n}^{-1}}y$

3. $\frac{1}{1+y^2}$

4. $\frac{1}{x(1+y^2)}$

Q. An integrating factor of the differential equation $x\frac{dy}{dx}+y\log x=x{e}^x{x}^{-\frac{1}{2}\log x},(x>0)$ is

1. $x^{\log x}$
2. $(\sqrt{x}{)}^{\log x}$
3. $(\sqrt{e}{)}^{\log x}$
4. $e^{x^2}$

Q. Let $y=y\left(t\right)$ be a solution of the differential equation $y^{\prime }+2ty=t^2$, then $16\underset{t\to \infty }{lim}\frac{y}{t}$ is

Q.

An integrating factor for the differential equation
$1+x^2)dx-(ta{n}^{-1}y-x)dy=0$

[MP PET 1993]

1. $ta{n}^{-1}y$

2. $e^{ta{n}^{-1}}y$

3. $\frac{1}{1+y^2}$

4. $\frac{1}{x(1+y^2)}$

Q. If $x=g\left(y\right)e^{f\left(y\right)}$ is a solution of the differential equation $(1+y^2)+(x-e^{-{\tan }^{-1}y})\frac{dy}{dx}=0$, satisfying the condition $x=0$ at $y=0$, then the value of $f\left(\sqrt{2}\right)+g\left(\sqrt{2}\right)$ is

Q. Consider the differential equation, $y^{2}dx+(x-\frac{1}{y})dy=0.$ If value of $y$ is $1$ when $x=1,$ then the value of $x$ for which $y=2,$ is:

1. $\frac{3}{2}-\sqrt{e}$
2. $\frac{5}{2}+\frac{1}{\sqrt{e}}$
3. $\frac{3}{2}-\frac{1}{\sqrt{e}}$
4. $\frac{1}{2}+\frac{1}{\sqrt{e}}$

Q. If integrating factor of $x(1-x^2)dy+(2x^{2}y-y-a{x}^3)dx=0$ is $e^{\int Pdx}$, then P is equal to

1. $\frac{2x^2-a{x}^3}{x(1-x^2)}$
   
2. $(2x^2-1)$
   
3. $\frac{2x^2-1}{a{x}^3}$
   
4. $\frac{(2x^2-1)}{x(1-x^2)}$