Linear Differential Equations of First Order

Q.

Let $y=t^{10}+1$ and $x=t^8+1,$ then $\frac{d^{2}y}{d{x}^2}$ is

1. $\frac{5t}{2}$

2. $20t^8$

3. $\frac{5}{16}{t}^6$

4. None of these.

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 $b$, ${\int }_1^{2}f\left(x\right)dx=\frac{62}{5}$

1. $5$

2. $\frac{62}{5}$

3. $\frac{31}{5}$

4. $10$

Q.

A present, a firm is manufacturing $1000$ items, It is estimated that the rate of change of production $P$ with respect to the 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 the item is

1. $2500$

2. $3000$

3. $3500$

4. $4500$

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. $10$
4. $\frac{31}{5}$

Q.

If $y\left(x\right)$ satisfies the differential equation $y’-y\tan x=2xsecx$ and $y\left(0\right)=0$, then:

1. $y\left(\frac{\mathrm{\pi }}{4}\right)=\frac{{\mathrm{\pi }}^2}{8\sqrt{2}}$

2. $y\left(\frac{\mathrm{\pi }}{4}\right)=\frac{{\mathrm{\pi }}^2}{18}$

3. $y\left(\frac{\mathrm{\pi }}{3}\right)=\frac{{\mathrm{\pi }}^2}{9}$

4. $y\left(\frac{\mathrm{\pi }}{3}\right)=\frac{4{\mathrm{\pi }}^2}{3}+\frac{2{\mathrm{\pi }}^2}{3\sqrt{2}}$

Q. Solution of the equation $co{s}^{2}x\frac{dy}{dx}-\left(tan2x\right)y=co{s}^{4}x,|x|<\frac{\pi }{4},$ where $\left(\frac{\pi }{6}\right)=\frac{3\sqrt{3}}{8}$ is

1. $y=tan2xco{s}^{2}x$
2. $y=cot2xco{s}^{2}x$
3. $y=\frac{1}{2}tan2xco{s}^{2}x$
4. $y=\frac{1}{2}cot2xco{s}^{2}x$

Q.

Let $f:R^{+}\to R^{+}$ be a differentiable function satisfying $f\left(xy\right)=\frac{f\left(x\right)}{y}+\frac{f\left(y\right)}{x}$ for all $x,yϵR^{+}$. Also $f\left(1\right)=0,f^{\prime }\left(1\right)=1$. If M is the greatest value of $f\left(x\right)$ then $[m+e]$ is ___ (where [.] represents Greatest Integer Function).

1. 3

2. 2

3. 0

4. 1

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 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. $2\log x=by+a$

Q. Is $y\frac{dy}{dx}=cosx$ a linear ODE?

1. True
2. False

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. 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. $\frac{ydy}{dx}+xy=x^2$ is a linear differential equation of first order.

1. True
2. False

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

Q. $\frac{ydy}{dx}+xy=x^2$ is a linear differential equation of first order.

1. True
2. False