Differential Equations · NDA previous year questions

Consider the following statements: 1. The magnitude of $\vec{a}\times \vec{b}$ is same as the area of a triangle with sides $\vec{a}$ and $\vec{b}$ 2. If $\vec{a}\times \vec{b}=\vec{0}$ where $\vec{a}\ne \vec{0},\vec{b}\ne \vec{0}$, then $\vec{a}=\lambda \vec{b}$ Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

The solution of the differential equation $\frac{dy}{dx}=\cos (y-x)+1$ is

1. A.$e^x[\sec (y-x)-\tan (y-x)]=c$
2. B.$e^x[\sec (y-x)+\tan (y-x)]=c$
3. C.$e^x\sec (y-x)\tan (y-x)=c$
4. D.$e^x=c\sec (y-x)\tan (y-x)$

If $y=a\cos 2x+b\sin 2x$, then

1. A.$\frac{d^{2}y}{d{x}^2}+y=0$
2. B.$\frac{d^{2}y}{d{x}^2}+2y=0$
3. C.$\frac{d^{2}y}{d{x}^2}-4y=0$
4. D.$\frac{d^{2}y}{d{x}^2}+4y=0$

The differential equation of the system of circles touching the y-axis at the origin is

1. A.$x^2+y^2-2xy\frac{dy}{dx}=0$
2. B.$x^2+y^2+2xy\frac{dy}{dx}=0$
3. C.$x^2-y^2+2xy\frac{dy}{dx}=0$
4. D.$x^2-y^2-2xy\frac{dy}{dx}=0$

Consider the following in respect of the differential equation : 1. The degree of the differential equation is 1. 2. The order of the differential equation is 2. Which of the above statements is/are correct ?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is the general solution of the differential equation $\frac{dy}{dx}+\frac{x}{y}=0$ ?

1. A.$x^2+y^2=c$
2. B.$x^2-y^2=c$
3. C.$x^2+y^2=cxy$
4. D.$x+y=c$

If $\vec{a}=\hat{i}-2\hat{j}+5\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-3\hat{k}$ then what is $(\vec{b}-\vec{a})\cdot (3\vec{a}+\vec{b})$ equal to?

1. A.106
2. B.$-106$
3. C.53
4. D.$-53$

The differential equation of the family of curves $y=p\cos (ax)+q\sin (ax)$, where p, q are arbitrary constants, is

1. A.$\frac{d^{2}y}{d{x}^2}-a^{2}y=0$
2. B.$\frac{d^{2}y}{d{x}^2}-ay=0$
3. C.$\frac{d^{2}y}{d{x}^2}+ay=0$
4. D.$\frac{d^{2}y}{d{x}^2}+a^{2}y=0$

The equation of the curve passing through the point $(-1,-2)$ which satisfies $\frac{dy}{dx}=-x^2-\frac{1}{x^3}$, is

1. A.$17x^{2}y-6x^2+3x^5-2=0$
2. B.$6x^{2}y+17x^2+2x^5-3=0$
3. C.$6xy-2x^2+17x^5+3=0$
4. D.$17x^{2}y+6xy-3x^5+5=0$

What is the order of the differential equation whose solution is $y=a\cos x+b\sin x+c{e}^{-x}+d$, where a, b, c and d are arbitrary constants?

1. A.$1$
2. B.$2$
3. C.$3$
4. D.$4$

What is the solution of the differential equation $\ln \left(\frac{dy}{dx}\right)=ax+by$? where c is an arbitrary constant.

1. A.$a{e}^{ax}+b{e}^{by}=c$
2. B.$\frac{1}{a}{e}^{ax}+\frac{1}{b}{e}^{by}=c$
3. C.$a{e}^{ax}+b{e}^{-by}=c$
4. D.$\frac{1}{a}{e}^{ax}+\frac{1}{b}{e}^{-by}=c$

1. A.Option A
2. B.Option B
3. C.Option C
4. D.Option D

Let $|\vec{a}|\ne 0,|\vec{b}|\ne 0$. $(\vec{a}+\vec{b}).(\vec{a}+\vec{b})=|\vec{a}{|}^2+|\vec{b}{|}^2$ holds if and only if

1. A.$\vec{a}$ and $\vec{b}$ are perpendicular
2. B.$\vec{a}$ and $\vec{b}$ are parallel
3. C.$\vec{a}$ and $\vec{b}$ are inclined at an angle of $45^{\circ }$
4. D.$\vec{a}$ and $\vec{b}$ are anti-parallel

If $\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$, then what is $\vec{r}\cdot (\hat{i}+\hat{j}+\hat{k})$ equal to?

1. A.$x$
2. B.$x+y$
3. C.$-(x+y+z)$
4. D.$(x+y+z)$

What is the solution of the differential equation $xdy-ydx=0$?

1. A.$xy=c$
2. B.$y=cx$
3. C.$x+y=c$
4. D.$x-y=c$

Which one of the following differential equations has a periodic solution? where $\mu >0$.

1. A.$\frac{d^{2}x}{d{t}^2}+\mu x=0$
2. B.$\frac{d^{2}x}{d{t}^2}-\mu x=0$
3. C.$x\frac{dx}{dt}+\mu t=0$
4. D.$\frac{dx}{dt}+\mu xt=0$

The order and degree of the differential equation $y^2=4a(x-a)$, where 'a' is an arbitrary constant, are respectively

1. A.$1,2$
2. B.$2,1$
3. C.$2,2$
4. D.$1,1$

What is the solution of $(1+2x)dy-(1-2y)dx=0$?

1. A.$x-y-2xy=c$
2. B.$y-x-2xy=c$
3. C.$y+x-2xy=c$
4. D.$x+y+2xy=c$

What are the order and degree, respectively, of the differential equation ${\left(\frac{d^{3}y}{d{x}^3}\right)}^2=y^4+{\left(\frac{dy}{dx}\right)}^5$?

1. A.4, 5
2. B.2, 3
3. C.3, 2
4. D.5, 4

The order and degree of the differential equation are respectively

1. A.3 and 2
2. B.2 and 2
3. C.2 and 3
4. D.1 and 3

The differential equation of minimum order by eliminating the arbitrary constants $A$ and $C$ in the equation $y=A[\sin (x+C)+\cos (x+C)]$ is

1. A.$y^{\prime \prime }+(\sin x+\cos x)y^{\prime }=1$
2. B.$y^{\prime \prime }=(\sin x+\cos x)y^{\prime }$
3. C.$y^{\prime \prime }=(y^{\prime }{)}^2+\sin x\cos x$
4. D.$y^{\prime \prime }+y=0$

The solution of the differential equation $\frac{dy}{dx}=\frac{y{\varphi }^{\prime }(x)-y^2}{\varphi (x)}$ is

1. A.$y=\frac{x}{\varphi (x)+c}$
2. B.$y=\frac{\varphi (x)}{x}+c$
3. C.$y=\frac{\varphi (x)+c}{x}$
4. D.$y=\frac{\varphi (x)}{x+c}$

What are the degree and order respectively of the differential equation $y=x{\left(\frac{dy}{dx}\right)}^2+{\left(\frac{dx}{dy}\right)}^2$ ?

1. A.1, 2
2. B.2, 1
3. C.1, 4
4. D.4, 1

What is the differential equation corresponding to $y^2-2ay+x^2=a^2$ by eliminating $a$? where $p=\frac{dy}{dx}$.

1. A.$(x^2-2y^2)p^2-4pxy-x^2=0$
2. B.$(x^2-2y^2)p^2+4pxy-x^2=0$
3. C.$(x^2+2y^2)p^2-4pxy-x^2=0$
4. D.$(x^2+2y^2)p^2-4pxy+x^2=0$

What is the general solution of the differential equation $ydx-(x+2y^2)dy=0$?

1. A.$x=y^2+cy$
2. B.$x=2c{y}^2$
3. C.$x=2y^2+cy$
4. D.None of the above