Solving Linear Differential Equations of First Order

Q.

If $R$ and $C$ denote the set of real numbers and complex numbers, respectively. Then, the function $f:C\to R$ defined by $f\left(z\right)=\left|z\right|$ is

1. one-one

2. onto

3. bijective

4. neither one-one nor onto

Q.

The value of $p$ for which the equation $x^2+pxy+y^2-5x-7y+6=0$ represents a pair of straight lines, is

1. $\frac{5}{2}$

2. $5$

3. $2$

4. $\frac{2}{5}$

Q. Let $y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}+2y=f\left(x\right),$ where $f\left(x\right)=\{\begin{array}{c}1,x\in [0,1]\\ 0,\text{otherwise}\end{array}$
If $y\left(0\right)=0$, then $y\left(\frac{3}{2}\right)$ is

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

Q. Let $y=y\left(x\right)$ be the solution of the differential equation, $(x^2+1{)}^2\frac{dy}{dx}+2x(x^2+1)y=1$ such that $y\left(0\right)=0$. If $\sqrt{a}y\left(1\right)=\frac{\pi }{32}$, then the value of $a$ is:

1. $\frac{1}{16}$
2. $\frac{1}{2}$
3. $1$
4. $\frac{1}{4}$

Q. If a curve passes through the point $(1,-2)$ and has slope of the tangent at any point $(x,y)$ on it as $\frac{x^2-2y}{x}$, then the curve also passes through the point :

1. $(3,0)$
2. $(-1,2)$
3. $(-\sqrt{2},1)$
4. $(\sqrt{3},0)$

Q. The equation $(x-a{)}^2+(y-b{)}^2=k(px+qy+r{)}^2,$ where $(pa+qb+r)\ne 0$ represents

1. a parabola for $k<\frac{1}{p^2+q^2}$
2. an ellipse for $0<k<\frac{1}{p^2+q^2}$
3. a hyperbola for $k>\frac{1}{p^2+q^2}$
4. a point circle for $k=0$

Q. A differentiable function $f$ satisfies $f\left(x\right)=\underset{0}{\overset{x}{\int }}\left(f\right(t)\cos t-\cos (t-x\left)\right)dt$. Which of the following hold(s) good?

1. $f\left(x\right)$ has the minimum value $1-e$
2. $f\left(x\right)$ has the maximum value $1-e^{-1}$
3. $f^{\prime \prime }\left(\frac{\pi }{2}\right)=e$
4. $f^{\prime }\left(0\right)=1$

Q. Let $f$ be a non-negative function defined on the interval $[0,\frac{\pi }{2}]$
If $\underset{0}{\overset{x}{\int }}\left(f^{{}^{\prime }}\right(t)-\sin 2t)dt=\underset{x}{\overset{0}{\int }}f\left(t\right)\tan tdt$
and $f\left(0\right)=1$, assume $y=f\left(x\right)$ then

1. $f\left(0\right)=1$ is the maximum value of $f$
2. $f\left(0\right)=1$ is the minimum value of $f$
3. $\underset{0}{\overset{\frac{\pi }{2}}{\int }}ydx=3-\frac{\pi }{2}$
4. $f\left(\frac{\pi }{3}\right)=1$

Q.

Which of the following equation is non-linear

1. $\frac{dy}{dx}=cosx$

2. $\frac{d^{2}y}{d{x}^2}+y=0$

3. $dx+dy=0$

4. $x\frac{dy}{dx}+\frac{3}{\frac{dy}{dx}}=y^2$

Q. Let, $f\left(x\right)=\{\begin{array}{c}\left[x\right],-2\le x\le -1\\ |x|+1,-1<x\le 2\end{array}$ and $g\left(x\right)=\{\begin{array}{c}\left[x\right],-\pi \le x\le 0\\ \sin x,0<x\le \pi \end{array},$ then the exhaustive domain of $g\left(f\right(x\left)\right)$ is

1. $[–2,0]$
2. $[–2,2]$
3. $[–1,2]$
4. $[0,2]$

Q. The solution of $(x+2y^3)\left(\frac{dy}{dx}\right)=y$ is (where c is arbitrary constant)

1. $x=y^3+cy$
2. $x=y^3-cy$
3. $y=y^3-cy$
4. $y=y^3+cy$

Q. If $y\left(x\right)$ is the solution of the differential equation
$(x+2)\frac{dy}{dx}=x^2+4x–9,x\ne –2$
$\text{and}y\left(0\right)=0,\text{then}y(–4)$is equal to :

1. $2$
2. $0$
3. $-1$
4. $1$

Q. Let $y=y\left(x\right)$ be the solution curve of the differential equation, $(y^2-x)\frac{dy}{dx}=1$, satisfying $y\left(0\right)=1$. This curve intersects the $x$-axis at a point whose abscissa is :

1. $2+e$
2. $2$
3. $2-e$
4. $-e$

Q. Let $\frac{dy}{dx}=\frac{y{\varphi }^{{}^{\prime }}\left(x\right)-y^2}{\varphi \left(x\right)}$, where $\varphi \left(x\right)$ is a specified function satisfying $\varphi \left(1\right)=1,\varphi \left(4\right)=1296$. If $y\left(1\right)=1$ then $\frac{1}{81}y\left(4\right)$ is equal to

Q. Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $x$ which satisfy ${\sin }^{-1}\left(\frac{3x}{5}\right)+{\sin }^{-1}\left(\frac{4x}{5}\right)={\sin }^{-1}x$ is equal to :

1. $0$
2. $3$
3. $2$
4. $1$

Q. If $y_1\left(x\right)$ is a solution of the differential equation $\frac{dy}{dx}-f\left(x\right)y=0$, then a solution of the differential equation $\frac{dy}{dx}-f\left(x\right)y=r\left(x\right)$ is

1. $\frac{1}{y_1\left(x\right)}\int r\left(x\right)y_1\left(x\right)dx$
2. $y_1\left(x\right)\int \frac{r\left(x\right)}{y_1\left(x\right)}dx$
3. $\int r\left(x\right)y_1\left(x\right)dx$
4. $noneofthese$

Q.

A curve passes through (2, 0) and the slope of tangent at a point P(x, y) is equal to $(x+1{)}^2+y-3/(x+1).$ Then equation of the curve is: [IIT 2004]

1. $y=x^2+2x$
2. $y=x^2-2x$
3. $y=2x^2-x$
4. None of these

Q. Solution of the differential equation
$cosxdy=y(sinx-y)dx,0<x<\frac{\pi }{2}$ is

1. $ysecx=tanx+c$
2. $ytanx=secx+c$
3. $tanx=(secx+c)y$
4. $secx=(tanx+c)y$

Q. Let $y=y\left(x\right)$ be the solution of the differential equation, $\frac{dy}{dx}+y\tan x=2x+x^2\tan x,x\in (-\frac{\pi }{2},\frac{\pi }{2})$, such that $y\left(0\right)=1.$ Then :

1. $y\left(\frac{\pi }{4}\right)-y(-\frac{\pi }{4})=\sqrt{2}$
2. $y\left(\frac{\pi }{4}\right)+y(-\frac{\pi }{4})=\frac{{\pi }^2}{2}+2$
3. $y^{\prime }\left(\frac{\pi }{4}\right)-y^{\prime }(-\frac{\pi }{4})=\pi -\sqrt{2}$
4. $y^{\prime }\left(\frac{\pi }{4}\right)+y^{\prime }(-\frac{\pi }{4})=-\sqrt{2}$

Q. A differentiable function $f$ satisfies $f\left(x\right)=\underset{0}{\overset{x}{\int }}\left(f\right(t)\cos t-\cos (t-x\left)\right)dt$. Which of the following hold(s) good?

1. $f\left(x\right)$ has the minimum value $1-e$
2. $f\left(x\right)$ has the maximum value $1-e^{-1}$
3. $f^{\prime \prime }\left(\frac{\pi }{2}\right)=e$
4. $f^{\prime }\left(0\right)=1$

Q. If $\cos x\frac{dy}{dx}+y\sin x=1$ and $y\left(\frac{\pi }{4}\right)=\sqrt{2}$, then $y\left(\frac{-\pi }{3}\right)$ is

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

Q. The solution of $(x+2y^3)\left(\frac{dy}{dx}\right)=y$ is (where c is arbitrary constant)

1. $x=y^3+cy$
2. $x=y^3-cy$
3. $y=y^3-cy$
4. $y=y^3+cy$