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Formation of differential equations

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Question

If x(t) is a solution of $(t + 1) d x = (2 x + {(t + 1)}^{4}) d t, x (0) = 2$ then $lim_{t \to 1} x (t)$ is

Moderate

Solution

$\frac{d x}{d t} - \frac{2}{t + 1} x = (t + 1)^{3}$ The I. F. is $e^{- 2 \int \frac{d t}{t + 1}} = \frac{1}{(t + 1)^{2}}$
The equation reduces to $\frac{d}{d t} (x \frac{1}{(t + 1)^{2}}) = (t + 1)$
$\Rightarrow \frac{x}{(t + 1)^{2}} = \frac{(t + 1)^{2}}{2} + C .$ Since $x (0) = h 2$
so , $C = \frac{3}{2}$ . Thus $\frac{x}{(t + 1)^{2}} = \frac{(t + 1)^{2}}{2} + \frac{3}{2}$
$\Rightarrow x = \frac{(t + 1)^{4}}{2} + \frac{3}{2} (t + 1)^{2}$ . $lim_{t \to 1} x (t) = 8 + 6 = 14$

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Similar Questions

Consider the family of curves $A x^{2} + B y^{2} = 1$ . If this family of curves is represented by the differential equation $P y " + Q {(y')}^{2} = y y'$

, where P and Q are functions of x and y, here $(y' = \frac{d y}{d x} a n d y " = \frac{d^{2} y}{d x^{2}})$ then which of the following statement(s) is/are true ?

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