Solving Linear Differential Equations of First Order

Q. If the solution of differential equation $x\ln x\frac{dy}{dx}+y=2\ln x$ is $y\left(\ln x\right)=(\ln x{)}^n+C$ then $n=$ ___

Q.

Find the value of$k$, if $x=2,y=1$ is a solution to the equation $2x+3y=k.$

Q.

If the system of linear equations

$$\begin{gathered} x+y+3z=0 \\ x+3y+k^{2}z=0 \\ 3x+y+3z=0 \end{gathered}$$

has a non-zero solution $(x,y,z)$for some $k\in R$, then $x+\left(\frac{y}{z}\right)$ is equal to:

1. $-9$

2. $9$

3. $-3$

4. $3$

Q.

Let $A=\left[\begin{array}{cc}i& -i\\ -i& i\end{array}\right]$ Then, the system of linear equations $A^8\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}8\\ 64\end{array}\right]$

1. No solution

2. Exactly two solutions

3. A unique solution

4. Infinitely many solutions

Q.

The differential coefficient of ${\text{x}}^6$ with respect to ${\mathrm{x}}^3$ is

1. $5{\mathrm{x}}^2$

2. $3{\mathrm{x}}^3$

3. $5{\mathrm{x}}^3$

4. $2{\mathrm{x}}^3$

Q. If $y=y\left(x\right)$ is the solution of the equation $e^{\sin y}\cos y\frac{dy}{dx}+e^{\sin y}\cos x=\cos x,$ $y\left(0\right)=0;$ then $1+y\left(\frac{\pi }{6}\right)+\frac{\sqrt{3}}{2}y\left(\frac{\pi }{3}\right)+\frac{1}{\sqrt{2}}y\left(\frac{\pi }{4}\right)$ is equal to

Q.

If ${\sec }^{-1}\left(\frac{1+x}{1-y}\right)=a$, then $\frac{dy}{dx}$ is equal to

1. $\frac{y-1}{x+1}$

2. $\frac{y+1}{x-1}$

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

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

Q. Let $f$ be a differentiable function with $\underset{x\to \infty }{lim}f\left(x\right)=0.$ If $y^{\prime }+y{f}^{\prime }\left(x\right)-f\left(x\right)f^{\prime }\left(x\right)=0,$ $\underset{x\to \infty }{lim}y\left(x\right)=0,$ then
$($where $y^{\prime }\equiv \frac{dy}{dx})$

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

Q. The area (in sq. units) bounded by the parabola $y=x^2-1$, the tangent at the point $(2,3)$ to it and the $y$-axis is:

1. $\frac{8}{3}$
2. $\frac{14}{3}$
3. $\frac{56}{3}$
4. $\frac{32}{3}$

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

Q.

If ${\cos }^{-1}\left(x\right)+{\cos }^{-1}\left(y\right)+{\cos }^{-1}\left(z\right)+{\cos }^{-1}\left(t\right)=4\pi$, then the value of $x^2+y^2+z^2+t^2$ is

1. $xy+zy+zt$

2. $1–2xyzt$

3. $4$

4. $6$

Q. The differential coefficient of ${\log }_{10}x$ with respect to ${\log }_{x}10$ is

1. $1$
2. $-({\log }_{10}x{)}^2$
3. $({\log }_{10}x{)}^2$
4. $\frac{x^2}{100}$

Q.

The equation of the circle circumscribing the triangle formed by the lines $x+y=6$, $2x+y=4$ and $x+2y=5$ is

1. $x^2+y^2+17x+19y-50=0$

2. $x^2+y^2-17x-19y-50=0$

3. $x^2+y^2+17x-19y-50=0$

4. $x^2+y^2-17x-19y+50=0$

Q.

A function $y=f\left(x\right)$ satisfies the condition $f^{\prime }\left(x\right)sinx+f\left(x\right)cosx=1,f\left(x\right)$ being bounded when $x\to 0$. if $I={\int }_0^{\frac{\pi }{2}}f\left(x\right)dx,$ then

1. $\frac{\pi }{2}<I<\frac{{\pi }^2}{4}$

2. $\frac{\pi }{4}<I<\frac{{\pi }^2}{2}$

3. $1<I<\frac{\pi }{2}$

4. $0<I<1$

Q.

The intercept on the line $y=x$ by the circle $x^2+y^2-2x=0$ is $AB$. Equation of the circle on $AB$ as a diameter is

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

2. $x\left(x-1\right)+y\left(y-1\right)=0$

3. $x^2+y^2=2$

4. $\left(x-1\right)\left(x-2\right)+\left(y-1\right)\left(y-2\right)=0$

Q.

Find the general solution of the Differential Equation $\frac{dy}{dx}-\frac{y}{x}=0.$

Q.

Find the first two derivatives of $2\sin x\cos x$.

Q.

If $\underset{x\to \infty }{lim}\left[\frac{x^3+1}{x^2+1}-(ax+b)\right]=2$, then

1. $a=1$ and $b=1$

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

3. $a=1$ and $b=-2$

4. $a=1$ and $b=2$

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

Q. The differential equation of family of circles touching $x-$axis at origin is

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

Q.

If $3\cos x\ne 2\sin x$, then the general solution of ${\sin }^{2}x-\cos 2x=2-\sin 2x$ is

1. $\frac{\mathrm{n\pi }+{(-1)}^{\mathrm{n}}\mathrm{\pi }}{2},n\in Z$

2. $\frac{n\pi }{2},n\in Z$

3. $\frac{\left(4\mathrm{n}\pm 1\right)\mathrm{\pi }}{2},n\in Z$

4. $(2n–1),n\in Z$

Q. The line $12x\cos \theta +5y\sin \theta =60$ is tangent to which of the following curves?

1. $25x^2+12y^2=3600$
2. $144x^2+25y^2=3600$
3. $x^2+y^2=169$
4. $x^2+y^2=60$

Q.

If the normal to the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ makes an angle $\phi$ with the $x-$ axis, then its equation is

1. $x\sin \left(\phi \right)+y\cos \left(\phi \right)=a$

2. $y\cos \left(\phi \right)-x\sin \left(\phi \right)=a\cos \left(2\phi \right)$

3. $x\cos \left(\phi \right)+y\sin \left(\phi \right)=a\sin \left(2\phi \right)$

4. None of these

Q. The function $y=f\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$ in (-1, 1) satisfying f(0) =0. Then ${\int }_{\frac{-\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}}f\left(x\right)d\left(x\right)$ is

1. $\frac{\pi }{3}-\frac{\sqrt{3}}{2}$
2. $\frac{\pi }{3}-\frac{\sqrt{3}}{4}$
3. $\frac{\pi }{6}-\frac{\sqrt{3}}{4}$
4. $\frac{\pi }{6}-\frac{\sqrt{3}}{2}$

Q. Consider the curve which satisfies the differential equation $(1+y^2)dx-xydy=0$ and also passes through the point (1, 0) If the point (a, b), a > 0, is the focus of the curve, then the value of is equal to

Q. The family of curves satisfying the differential equation $\frac{dy}{dx}+\frac{1}{x}\sin 2y=x^3{\cos }^{2}y,$ is
(where $C$ is an arbitrary constant)

1. $x^6+6x^2=C\tan y$
2. $6x^2\tan y=x^6+C$
3. $\sin 2y=x^3{\cos }^{2}y+C$
4. $y^6=6y^2\tan x+C$

Q.

The solution of the equation $\frac{dy}{dx}+ytanx=x^{m}cosx$ is

1. $(m+1)y=x^{m+1}cosx+c(m+1)cosx$
2. $(my=(x^m+c)cosx$
3. $y=(x^{m+1}+c)cosx$

4. $Noneofthese$

Q. Which of the following functions is periodic

1. $\text{sgn}\left(e^{-x}\right)$
2. $\sin x+|\sin x|$
3. $min.\{\sin x,|x|\}$
4. $[x+\frac{1}{2}]+[x-\frac{1}{2}]+2[-x]$
   (where [.] denotes greatest integer function)

Q.

The differential equation representing the family of curves $y^2=2c(x+\sqrt{c})$, where $c>0$ is a parameter is of order and degree as follows,

1. order $2$, degree $2$

2. order $1$, degree $3$

3. order $1$, degree $1$

4. order $1$, degree $2$

Q.

Use the graph to write a linear function that relates $\mathrm{y}$ to $\mathrm{x}$.