SATHEE: Differential Equations 2 Question 23

Differential Equations 2 Question 23

24. Let $u (x)$ and $v (x)$ satisfy the differential equations $\frac{d u}{d x} + p (x) u = f (x)$ and $\frac{d v}{d x} + p (x) v = g (x),$ where $p (x), f (x)$ and $g (x)$ are continuous functions. If $u (x_{1}) > v (x_{1})$ for some $x_{1}$ and $f (x) > g (x)$ for all $x > x_{1}$ , prove that any point $(x, y)$ where $x > x_{1}$ does not satisfy the equations $y = u (x)$ and $y = v (x)$ .

$(1997, 5$ M)

Integer Answer Type Question

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Answer:

Correct Answer: 24. (0)

Solution:

$w (x) = u (x) - v (x)$

and

$h (x) = f (x) - g (x)$

On differentiating Eq. (i) w.r.t. $x$

$\frac{d w}{d x} = \frac{d u}{d x} - \frac{d v}{d x}$

$= f (x) - p (x) \cdot u (x) - g (x) - p (x) v (x)$ [given]

$= f (x) - g (x) - p (x) [u (x) - v (x)]$

$\Rightarrow \frac{d w}{d x} = h (x) - p (x) \cdot w (x)$

$\Rightarrow \frac{d w}{d x} + p (x) w (x) = h (x)$ which is linear differential equation

The integrating factor is given by

$I F = e^{\int p (x) d x} = r (x)$

On multiplying both sides of Eq. (ii) of $r (x)$ , we get

$\begin{aligned} r (x) \cdot \frac{d w}{d x} + p (x) (r (x)) w (x) = r (x) \cdot h (x) \\ \Rightarrow & \frac{d}{d x} [r (x) w (x)] = r (x) \cdot h (x) ∵ \frac{d r}{d x} = p (x) \cdot r (x) \end{aligned}$

Now,

$r (x) = e^{\int P (x) d x} > 0, \forall x$

and

$h (x) = f (x) - g (x) > 0, for x > x_{1}$

Thus,

$\frac{d}{d x} [r (x) w (x)] > 0, \forall x > x_{1}$

$r (x) w (x)$ increases on the interval $[x, \infty$ [

Therefore, for all $x > x_{1}$

$\begin{aligned} r (x) w (x) & > r (x_{1}) w (x_{1}) > 0 \\ [∵ r (x_{1}) > 0 and u (x_{1}) > v (x_{1})] \\ w (x) & > 0 \forall x > x_{1} \\ u (x) > v (x) \forall x > x_{1} & [∵ r (x) > 0] \end{aligned}$

$\begin{array}{ll} \Rightarrow & w (x) > 0 \forall x > x_{1} \\ \Rightarrow & u (x) > v (x) \forall x > x_{1} \end{array}$

Hence, there cannot exist a point $(x, y)$ such that $x > x_{1}$ and $y = u (x)$ and $y = v (x)$ .

Differential Equations 2 Question 23

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Differential Equations 2 Question 23

24. Let u(x) and v(x) satisfy the differential equations dudx+p(x)u=f(x) and dvdx+p(x)v=g(x), where p(x),f(x) and g(x) are continuous functions. If u(x1)>v(x1) for some x1 and f(x)>g(x) for all x>x1, prove that any point (x,y) where x>x1 does not satisfy the equations y=u(x) and y=v(x).

Integer Answer Type Question

Answer:

Engineering Preparation

Medical Preparation

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Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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