Degree of a Differential Equations

Q. The degree and order of the differential equation of the family of all parabolas whose axis is x–axis, are respectively

1. 1, 2
2. 3, 2
3. 2, 3
4. 2, 1

Q.

The product of two odd number is:

1. an even number

2. an odd number

3. either even or odd

4. prime number

Q.

If $f$ be a polynomial, then the second derivative of $f\left(e^x\right)$ is

1. $f\left(e^x\right)$

2. $f\left(e^x\right)e^x+f\left(e^x\right)$

3. $f\left(e^x\right)e^{2x}+f\left(e^x\right)$

4. $f\left(e^x\right)e^{2x}+f\left(e^x\right)e^x$

Q.

The order of the differential equation of a family of curves represented by an equation containing four arbitrary constants, will be

1. $2$

2. $4$

3. $6$

4. None of these

Q.

Find the degree measure of -4

Q.

Family $y=Ax+A^3$ of curves represented by the differential equation of degree

1. Three

2. Two

3. One

4. None of these

Q.

The order and degree of the differential equation ${[4+{\left(\frac{dy}{dx}\right)}^2]}^{2/3}=\frac{d^{2}y}{d{x}^2}$ are

1. 2, 2

2. 3, 3

3. 2, 3

4. 3, 2

Q.

Find the differential equation of all non-vertical lines in a plane.

Q. Evaluate: $\int \frac{x}{x^4-1}dx$

Q. If p and q are order and degree of differential equation $y^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2+3x{\left(\frac{dy}{dx}\right)}^{\frac{1}{3}}+x^2{y}^2=sinx$, then

1. $\frac{p}{q}=\frac{1}{2}$
2. p = q
3. p < q
4. p > q

Q. If both f(x) and g(x) are differentiable functions at $x=x_0$, then the function defined as h(x) = maximum {f(x), g(x)} :

1. Is never differentiable at
2. Is always differentiable at
3. Is differentiable at provided
4. Cannot be differentiable at if

Q. The order and degree of the differential equation ${[1+{\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)]}^{3/4}=\frac{d^{2}y}{d{x}^2}$

1. order $=2$, degree $=3$
2. order $=2$, degree $=4$
3. order $=2$, degree $=\frac{3}{4}$
4. order $=2$, degree $=$ not defined

Q. The differential equation(s) of family of curves whose tangent form an angle of $\frac{\pi }{4}$ with the hyperbola $xy=c^2$ is/are given by

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

Q. Degree of the differential equation
$\frac{d^{4}y}{d{x}^4}+(\frac{d^{3}y}{d{x}^3}{)}^2+(\frac{d^{2}y}{d{x}^2}{)}^3+(\frac{dy}{dx}{)}^4=0$ is ___

Q. The solution of $\frac{dy}{dx}=e^{3x+4y}$ with $y\left(0\right)=0$ is $4e^{3x}+3e^{-4y}=\alpha ,$ then the value of $\alpha$ is

Q. Integrate the rational function: $\frac{2x}{(x^2+1)(x^2+3)}$

Q. The order and degree of the differential equation ${(1+3\frac{dy}{dx})}^{\frac{2}{3}}=4\frac{d^{3}y}{d{x}^3}$ are

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

Q.

The differential equation corresponding to primitive $y=e^{dx}$ is

or

The elimination of the arbitrary constant m from the equation $y=e^{mx}$ gives the differential equation

[MP PET 1995, 2000; Pb. CET 2000]

1. 
   

2. 
   

3. 
   

4. 
   

Q. Prove ${\int }_0^{1}x{e}^{x}dx=1$

Q. The inverse of the bijective function $f\left(x\right)=2\sqrt[3]{3x+1}-3,$ is

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

Q.

The order and degree of the differential equation $y=x\frac{dy}{dx}+\sqrt{a^2{\left(\frac{dy}{dx}\right)}^2+b^2}$ are

1. 1, 2

2. 2, 1

3. 1, 1

4. 2, 2

Q.

Is every linear equation is linear function ?

Q. Let $f\left(x\right),g\left(x\right)$ and $h\left(x\right)$ be polynomials of degree $4$ and satisfy the equation $\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ a& b& c\\ p& q& r\end{array}\right|=l{x}^4+m{x}^3+n{x}^2+4x+1$ where $a,b,c,p,q,r,l,m,n$ are real constants. Then the value of $\left|\begin{array}{ccc}{f}^{\prime \prime \prime }\left(0\right)-f^{\prime \prime }\left(0\right)& g^{\prime \prime \prime }\left(0\right)-g^{\prime \prime }\left(0\right)& h^{\prime \prime \prime }\left(0\right)-h^{\prime \prime }\left(0\right)\\ a& b& c\\ p& q& r\end{array}\right|$ is

1. $3m-n$
2. $2(3m-n)$
3. $3(3m+n)$
4. $3m+n$

Q.

The solution of the equation $\frac{dy}{dx}-e^{x-y}+x^2{e}^{-y}$ is
[MP PET 2004]

1. 
   

2. 
   

3. 
   

4. 
   

Q. Evaluate : $\int \frac{x}{\sqrt{x}+1}dx$

Q. Explain slope of a line with an example.

Q. Integrate the function:
$f^{\prime }(ax+b)\left[f\right(ax+b){]}^n$

Q. The differential equation of the family of curves $y=e^x(A\cos x+B\sin x)$ where $A,B$ are arbitary constants is

1. $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$
2. $\frac{d^{2}y}{d{x}^2}-9x=13$
3. $\frac{d^{2}y}{d{x}^2}+3y=4$
4. ${\left(\frac{dy}{dx}\right)}^2+\frac{dy}{dx}-xy=0$

Q. Integrate the function: $\frac{x+2}{\sqrt{x^2+2x+3}}$

Q. The integrating factor of the differential equation $(x+y^2)\frac{dy}{dx}=1$ is

1. $e^{-y}$
2. $e^{-x}$
3. $e^x$
4. $e^y$