Formation of differential equations — practice questions

Question 1 · difficulty L3 · understanding

If $${y^{1/4}} + {y^{ - 1/4}} = 2x$$, and $$({x^2} - 1){{{d^2}y} \over {d{x^2}}} + \alpha x{{dy} \over {dx}} + \beta y = 0$$, then | $$\alpha$$ $$-$$ $$\beta$$ | is equal to __________.

Question 2 · difficulty L3 · understanding

Let a curve y = f(x) pass through the point (2, (log e 2) 2 ) and have slope $${{2y} \over {x{{\log }_e}x}}$$ for all positive real value of x. Then the value of f(e) is equal to ______________.

Question 3 · difficulty L3 · understanding

Let $$f(x)$$ be a positive function such that the area bounded by $$y=f(x), y=0$$ from $$x=0$$ to $$x=a>0$$ is $$e^{-a}+4 a^2+a-1$$. Then the differential equation, whose general solution is $$y=c_1 f(x)+c_2$$, where $$c_1$$ and $$c_2$$ are arbitrary constants, is

• A. $$\left(8 e^x+1\right) \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$$
• B. $$\left(8 e^x+1\right) \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$$
• C. $$\left(8 e^x-1\right) \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$$
• D. $$\left(8 e^x-1\right) \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$$

Question 4 · difficulty L3 · understanding

The differential equation of the family of circles passing through the origin and having centre at the line $$y=x$$ is :

• A. $$\left(x^2-y^2+2 x y\right) \mathrm{d} x=\left(x^2-y^2+2 x y\right) \mathrm{d} y$$
• B. $$\left(x^2+y^2-2 x y\right) \mathrm{d} x=\left(x^2+y^2+2 x y\right) \mathrm{d} y$$
• C. $$\left(x^2+y^2+2 x y\right) \mathrm{d} x=\left(x^2+y^2-2 x y\right) \mathrm{d} y$$
• D. $$\left(x^2-y^2+2 x y\right) \mathrm{d} x=\left(x^2-y^2-2 x y\right) \mathrm{d} y$$

Question 5 · difficulty L3 · understanding

Let a differentiable function $$f$$ satisfy f(x)+\int_\limits{3}^{x} \frac{f(t)}{t} d t=\sqrt{x+1}, x \geq 3. Then $$12 f(8)$$ is equal to :

• A. 19
• B. 34
• C. 17
• D. 1

Question 6 · difficulty L3 · understanding

The differential equation of the family of circles passing through the points $$(0,2)$$ and $$(0,-2)$$ is :

• A. $$2 x y \frac{d y}{d x}+\left(x^{2}-y^{2}+4\right)=0$$
• B. $$2 x y \frac{d y}{d x}+\left(x^{2}+y^{2}-4\right)=0$$
• C. $$2 x y \frac{d y}{d x}+\left(y^{2}-x^{2}+4\right)=0$$
• D. $$2 x y \frac{d y}{d x}-\left(x^{2}-y^{2}+4\right)=0$$

Question 7 · difficulty L3 · understanding

Let a smooth curve $$y=f(x)$$ be such that the slope of the tangent at any point $$(x, y)$$ on it is directly proportional to $$\left(\frac{-y}{x}\right)$$. If the curve passes through the points $$(1,2)$$ and $$(8,1)$$, then $$\left|y\left(\frac{1}{8}\right)\right|$$ is equal to

• A. $$2 \log _{e} 2$$
• B. 4
• C. 1
• D. $$4 \log _{e} 2$$

Question 8 · difficulty L3 · understanding

Let $${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R$$, represents a circle with center ($$\alpha$$, $$\beta$$). Then, $$\alpha$$ + 2$$\beta$$ is equal to :

• A. $$-$$1
• B. 0
• C. 1
• D. 2

Question 9 · difficulty L3 · understanding

Let $${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}}$$, where a, b, c are constants, represent a circle passing through the point (2, 5). Then the shortest distance of the point (11, 6) from this circle is :

• A. 10
• B. 8
• C. 7
• D. 5

Question 10 · difficulty L3 · understanding

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point (2, $$-$$3) from the line 3x + 4y = 5, is given by :

• A. $$10{{{d^2}y} \over {d{x^2}}} = 11$$
• B. $$11{{{d^2}x} \over {d{y^2}}} = 10$$
• C. $$10{{{d^2}x} \over {d{y^2}}} = 11$$
• D. $$11{{{d^2}y} \over {d{x^2}}} = 10$$