Solving Homogeneous Differential Equations

Q.

If $2^x+2^y=2^{x+y},$ then $\frac{dy}{dx}$ is equal to

1. $\frac{(2^x+2^y)}{(2^x–2^y)}$

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

3. $\frac{2^{x-y}(2^y-1)}{(1–2^x)}$

4. None of these

Q.

Solve the differential equation $\frac{dy}{dx}=1+x+y^2+x{y}^2,$ when y=0 and x=0.

Q.

The number of values of $k$ for which the system of equations $(k+1)x+8y=4k;kx+(k+3)y=3k–1$ has infinitely many solutions is

1. $0$

2. $1$

3. $2$

4. Infinite

Q.

The intersection of three lines $x-y=0$, $x+2y=3$ and $2x+y=6$ is a

1. Equilateral triangle.

2. Right-angled triangle.

3. Isosceles triangle.

4. None of the above.

Q.

The general solution of the differential equation $\frac{dy}{dx}=\frac{x+y+1}{2x+2y+1}$ is

1. $lo{g}_e|3x+3y+2|+3x+6y=C$

2. ${\log }_e|3x+3y+2|-3x+6y=C$

3. ${\log }_e|3x+3y+2|-3x-6y=C$

4. ${\log }_e|3x+3y+2|+3x-6y=C$

Q.

If$S\mathrm{in}\left(xy\right)+\frac{x}{y}=x^2-y$ then $\frac{dy}{dx}$is equal to

1. $\frac{y(2xy-x^2\cos xy-1)}{(x{y}^2\cos xy–x+y^2)}$

2. $\frac{(2xy-y^2\cos xy-1)}{(x{y}^2\cos xy–x+y^2)}$

3. $\frac{y(2xy-y^2\cos xy-1)}{(x{y}^2\cos xy–x+y^2)}$

4. None of these

Q.

If $2A=3B$ and $4B=5C$ then $A:C=?$

1. $8:15$

2. $15:8$

3. $4:5$

4. $3:4$

Q.

The solution of the differential equation $9y\left(\frac{dy}{dx}\right)+4x=0$ is

1. $\left(\frac{y^2}{9}\right)+\left(\frac{x^2}{4}\right)=c$

2. $\left(\frac{y^2}{4}\right)+\left(\frac{x^2}{9}\right)=c$

3. $\left(\frac{y^2}{9}\right)-\left(\frac{x^2}{4}\right)=c$

4. ${\left(y\right)}^2+\left(\frac{x^2}{9}\right)=c$

Q.

The function $\frac{e^{2x}-1}{e^{2x}+1}$ is

1. Increasing

2. Decreasing

3. Even

4. Strictly increasing

Q.

Evaluate $\left(\sec 2\mathrm{A}+1\right){\sec }^2\mathrm{A}=$

1. $\sec \mathrm{A}$

2. $2\sec \mathrm{A}$

3. $\sec 2\mathrm{A}$

4. $2\sec 2\mathrm{A}$

Q.

The equations of the lines through $\left(1,1\right)$ and making angles of $45°$ with the line $x+y=0$ are

1. $x-1=0,x-y=0$

2. $x-y=0,y-1=0$

3. $x+y-2=0,y-1=0$

4. $x-1=0,y-1=0$

Q.

If $f\left(x\right)=\frac{x+2}{3x-1}$, then $f\left(f\left(x\right)\right)$ is equal to

1. $x$

2. $-x$

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

4. $-\frac{1}{x}$

5. $0$

Q.

$$\begin{gathered} 9x+3y+12=0 \\ 18x+6y+24=0 \end{gathered}$$

Q.

Integrate $\int \frac{1}{1+cotx}dx$ using substitution method.

Q.

The solution of the differential equation $\frac{dy}{dx}=\frac{x-y+3}{2(x-y)+5}$ is

1. $2(x-y)+\log (x-y)=x+c$

2. $2(x-y)-\log (x-y+2)=x+c$

3. $2(x-y)+\log (x-y+2)=x+c$

4. None of these

Q.

The equation $2{\cos }^{-1}x+{\sin }^{-1}x=\frac{11\mathrm{\pi }}{6}$ has.

1. No solution

2. Only one solution

3. Two solutions

4. Three solutions

Q.

The equation of the curve through point $\left(1,0\right)$ which satisfies the differential equation $\left(1+{\mathrm{y}}^2\right)d\mathrm{x}-\mathrm{xy}d\mathrm{y}=0$ is

1. ${\mathrm{x}}^2+{\mathrm{y}}^2=4$

2. ${\mathrm{x}}^2-{\mathrm{y}}^2=1$

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

4. None of these

Q.

The solution set of the equation $\left({\log }_{x}2\right)\left({\log }_{2x}2\right)=\left({\log }_{4x}2\right)$ is

1. $\{2^{-\sqrt{2}},2^{\sqrt{2}}\}$

2. $\left\{\frac{1}{3},3\right\}$

3. $\left\{\frac{1}{4},4\right\}$

4. None of these

Q. A solution of the differential equation $(x^2+xy+4x+2y+4)\frac{dy}{dx}-y^2=0,x>0$, passes through the point $(1,3)$. Then the solution curve:

1. intersects $y=x+2$ exactly at one point
2. intersects $y=x+2$ exactly at two points
3. intersects $y=(x+2{)}^2$
4. does NOT intersect $y=(x+3{)}^2$

Q. The solution of $(y+x+5)dy=(y-x+1)dx$ is

1. $log\left(\right(y+3{)}^2+(x+2{)}^2)+ta{n}^{-1}\frac{y+3}{x+2}=C$
2. $log\left(\right(y+3{)}^2+(x-2{)}^2)+ta{n}^{-1}\frac{y-3}{x-2}=C$
3. $log\left(\right(y+3{)}^2+(x+2{)}^2)+2ta{n}^{-1}\frac{y+3}{x+2}=C$
4. $log\left(\right(y+3{)}^2+(x+2{)}^2)-2ta{n}^{-1}\frac{y+3}{x+2}=C$

Q. The curve amongst the family of curves represented by the differential equation, $(x^2-y^2)dx+2xydy=0$ which passes through $(1,1)$, is :

1. a circle with centre on the y-axis
2. a circle with centre on the x-axis
3. an ellipse with major axis along the y-axis
4. a hyperbola with transverse axis along the x-axis

Q.

If the curves, $\frac{x^2}{a}+\frac{y^2}{b}=1$ and $\frac{x^2}{c}+\frac{y^2}{d}=1$ intersect each other at an angle of $90°$, then which of the following relations is true?

1. $a+b=c+d$

2. $a-b=c-d$

3. $ab=\frac{c+d}{a+b}$

4. $a-c=b+d$

Q. The solution of $\frac{dy}{dx}=\frac{y}{x}+tan\frac{y}{x}$ is

1. $ysin\left(\frac{y}{x}\right)=cx$
2. $ysin\left(\frac{y}{x}\right)=cy$
3. $sin\left(\frac{y}{x}\right)=cx$
4. $sin\left(\frac{x}{y}\right)=cx$

Q. Let $f$ be a real-valued differentiable function on $R$ (the set of all real numbers) such that $f\left(1\right)=1$. If the $y-$ intercept of the tangent at any point $P(x,y)$ on the curve $y=f\left(x\right)$ is equal to the cube of the abscissa of $P$, then the value of $f(–3)$ is equal to

Q.

Find a particular solution of the differential equation $(x-y)(dx+dy)=dx-dy$ given that y = -1, when x = 0.

Q. If the solution of the differential equation $x\left(\frac{d^{3}y}{d{x}^3}\right)+\frac{d^{2}y}{d{x}^2}+2x\left(\frac{dy}{dx}\right)-y+1=0$ is a straight line of the form $y=mx+c,$ where $m,c$ are real constants, then the number of such possible lines is

1. $1$
2. $2$
3. $3$
4. infinite

Q. Solution of the differential equation : $\frac{dy}{dx}=\frac{3x^2{y}^4+2xy}{x^2-2x^3{y}^3}$ is

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

Q.

If the lines $3x+4y+1=0,5x+\lambda y+3=0$ and $2x+y-1=0$ are concurrent, then $\lambda$ is equal to:

1. $-8$

2. $8$

3. $4$

4. $-4$

Q.

The tangent to the curve $y=a{x}^2+bx$ at $\left(2,-8\right)$ is parallel to $X$-axis. Then

1. $a=2,b=-2$

2. $a=2,b=-4$

3. $a=2,b=-8$

4. $a=4,b=-4$

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 $\left|y\left(1\right)\right|$ is equal to