Linear Differential Equations of First Order

Q.

Solve the following equation and check your results: $2y+\frac{5}{3}=\frac{26}{3}-y$

Q.

If $y=\frac{1}{(a-z)}$, then $\frac{dz}{dy}=$

1. ${\left(a-z\right)}^2$

2. $-{\left(z-a\right)}^2$

3. ${\left(z+a\right)}^2$

4. $-{\left(z+a\right)}^2$

Q.

If ${\cos }^{-1}\left(\frac{2}{3x}\right)+{\cos }^{-1}\left(\frac{3}{4x}\right)=\frac{\pi }{2}$, $x>\frac{3}{4}$ then $x=?$

1. $\frac{\sqrt{145}}{11}$

2. $\frac{\sqrt{145}}{12}$

3. $\frac{\sqrt{145}}{13}$

4. $\frac{\sqrt{145}}{14}$

Q. Let $y=y\left(x\right)$ be the solution of the differential equation, $x\frac{dy}{dx}+y=x{\log }_{e}x,(x>1).$ If $2y\left(2\right)={\log }_{e}4-1,$ then $y\left(e\right)$ is equal to:

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

Q.

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

1. $y=C\left(x+a\right)\left(1-ay\right)$

2. $y=C\left(x+a\right)\left(1+ay\right)$

3. $y=C\left(x-a\right)\left(1+ay\right)$

4. None of these.

Q.

Simplify and solve the following linear equation: $15(y-4)-2(y-9)+5(y+6)=0$.

Q.

Solve $\left(\frac{dy}{dx}\right)+\frac{y}{x}=x^3$:

1. $y=\frac{x^4}{5}+\frac{c}{x}$

2. $y=\frac{x^3}{3}+\frac{c}{x}$

3. $y=\frac{x^3}{3}+c$

4. $y=\frac{x^4}{4}+c$

Q. Let $y\left(x\right)$ be the solution of the differential equation $2x^{2}dy+(e^y-2x)dx=0,x>0.$ If $y\left(e\right)=1$, then $y\left(1\right)$ is equal to

1. $0$
2. $2$
3. ${\log }_e\left(2e\right)$
4. ${\log }_e\left(2\right)$

Q. If a curve $y=f\left(x\right)$ passes through the point $(1,2)$ and satisfies $x\frac{dy}{dx}+y=b{x}^4,$ then for what value of $b,$ $\underset{1}{\overset{2}{\int }}f\left(x\right)dx=\frac{62}{5}$ ?

1. $5$
2. $\frac{62}{5}$
3. $\frac{31}{5}$
4. $10$

Q. The equation $\frac{x^2}{10-a}+\frac{y^2}{4-a}=1$ represents an ellipse if

1. 4 < a < 10
2. a > 4
3. a > 10
4. a < 4

Q. Let $y=y\left(x\right)$ be the solution of the differential equation $\left(\right(x+2)e^{\left(\frac{y+1}{x+2}\right)}+(y+1\left)\right)dx=(x+2)dy,y\left(1\right)=1.$ If the domain of $y=y\left(x\right)$ is an open interval $(\alpha ,\beta ),$ then $|\alpha +\beta |$ is equal to

Q.

The integrating factor of the differential equation $\left(y\log y\right)dx=\left[\log \left(y\right)-x\right]dy$ is

1. $\frac{1}{\log y}$

2. $\log \left(\log y\right)$

3. $1+\log \left(y\right)$

4. $\frac{1}{\log \left(\log y\right)}$

5. $\log \left(y\right)$

Q. Let $\frac{dy}{dx}+y=f\left(x\right)$ where $y$ is a continuous function of $x$ with $y\left(0\right)=1$ and $f\left(x\right)=\{\begin{array}{cc}{e}^{-x},& 0\le x\le 2\\ e^{-2},& x>2\end{array}$.
Which of the following hold(s) good?

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

Q.

If $\mathrm{y}={\left(\frac{\mathrm{x}}{\mathrm{n}}\right)}^{\mathrm{nx}}\left(1+\log \left(\frac{\mathrm{x}}{\mathrm{n}}\right)\right)$, then the value of $\mathrm{y}\left(\mathrm{n}\right)$ is given by

1. $\frac{1}{\mathrm{n}}$

2. ${\left(\frac{1}{\mathrm{n}}\right)}^{\mathrm{n}}$

3. $\frac{\left[{\mathrm{n}}^2+1\right]}{\mathrm{n}}$

4. ${\left(\frac{1}{\mathrm{n}}\right)}^{\mathrm{n}}\left(\frac{\left[{\mathrm{n}}^2+1\right]}{\mathrm{n}}\right)$

Q.

The solution of the differential equation $\left(x^2-y^2\right)dx+2xydy=0$ is

1. $x^2+y^2=Cy$

2. $C\left(x^2-y^2\right)=x$

3. $x^2-y^2=Cy$

4. $x^2+y^2=Cx$

Q.

The Integrating Factor (IF) of the differential equation $x\frac{dy}{dx}-y=2x^2$ is
(a)$e^{-x}$
(b)$e^{-y}$
(c)$\frac{1}{x}$
(d)x

Q. At present, a firm is manufacturing $2000$ items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx}=100-12\sqrt{x}$. If the firm employs $25$ more workers, then the new level of production of items is :

1. $2500$
2. $3000$
3. $3500$
4. $4000$

Q. If the curve $y=y\left(x\right)$ represented by the solution of the differential equation $(2x{y}^2-y)dx+xdy=0$, passes through the intersection of the lines, $2x-3y=1$ and $3x+2y=8$, then $|y\left(1\right)|$ is equal to

Q.

The integrating factor of the differential equation $x\log \left(x\right)\frac{dy}{dx}+y=2\log \left(x\right)$ is given by

1. $e^x$

2. $\log \left(x\right)$

3. $\log \left(\log x\right)$

4. $x$

Q.

The solution $\left(1+x^2\right)\frac{dy}{dx}+2xy-4x^2=0$ is

1. $3x(1+y^2)=4y^3+c$

2. $3y(1+x^2)=4x^3+c$

3. $3x(1-y^2)=4y^3+c$

4. $3y(1+y^2)=4x^3+c$

Q. Solution of the differential equation $\frac{dy}{dx}+2y=\cos x$ is:
(where $C$ is integration constant)

1. $y=\frac{1}{5}[\cos x+2\sin x]+C{e}^{-2x}$
2. $y=\frac{4}{5}[2\cos x-\sin x]+C{e}^{-2x}$
3. $y=\frac{1}{5}[\cos x-\sin x]+C{e}^{-2x}$
4. $y=\frac{1}{5}[2\cos x+\sin x]+C{e}^{-2x}$

Q. Find the particular solution of the differential equation $(1+x^2)\frac{dy}{dx}=(e^{m{\tan }^{-1}x}-y)$, given that $y=1$, when $x=0$.

Q.

What is the solution of the equation $x-1=0$?

Q.

If $y=x+e^x$, then $\frac{{\mathrm{d}}^{2}y}{d{x}^2}$ is

1. $e^x$

2. $-\frac{e^x}{{\left(1+e^x\right)}^3}$

3. $-\frac{e^x}{{\left(1+e^x\right)}^2}$

4. $-\frac{1}{{\left(1+e^x\right)}^3}$

Q.

The differential equation whose solution is ${\mathrm{Ax}}^2+{\mathrm{By}}^2=1$, where $\mathrm{A}$ and $\mathrm{B}$ are arbitrary constant is of:

1. first-order and second degree

2. first order and first degree

3. second-order and first degree

4. second-order and second degree

Q. The differential equation $\frac{dy}{dx}=\frac{\sqrt{1-y^2}}{y}$ determines a part of circle with

1. Variable radii and fixed centre at $(0,1)$
2. Variable radii and fixed centre at $(0,-1)$
3. Fixed radius $1$ and variable center along the $x-$axis.
4. Fixed radius $1$ and variable center along the $y-$axis.

Q.

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

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

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

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

4. $1$

Q.

Show that the given differential equation is homogeneous and then solve it.

$xdy-ydx=\sqrt{x^2+y^2}dx$

Q.

Eccentricity of the ellipse $x^2+2y^2-2x+3y+2=0$ is

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

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

3. $\frac{1}{2\sqrt{2}}$

4. $\frac{1}{\sqrt{3}}$

Q. The equation of the curve which is such that the portion of the x-axis cut between the origin and tangent at any point is proportional to the ordinate of that point is (where $b$ is a constant of proportionality)

1. $x=y(a-b\log y)$
2. $\log x=b{y}^2+a$
3. $x^2=y(a-b\log y)$
4. $2\log x=by+a$