If for x $\ge$ 0, y = y(x) is the solution of the
differential equation
(x + 1)dy = ((x + 1)2 + y – 3)dx, y(2) = 0,
then y(3) is equal to _______.
Answer (integer)
3
Solution
(x + 1)dy = ((x + 1)<sup>2</sup> + y – 3)dx
<br><br>$\Rightarrow$ (1 + x)${{dy} \over {dx}}$ - y = (1 + x)<sup>2</sup> - 3
<br><br>$\Rightarrow$ $${{dy} \over {dx}} - {y \over {1 + x}} = \left( {1 + x} \right) - {3 \over {1 + x}}$$
<br><br>I.F = ${e^{ - \int {{{dx} \over {1 + x}}} }}$ = ${1 \over {1 + x}}$
<br><br>Solution of the differential equation,
<br><br>$y\left( {{1 \over {1 + x}}} \right)$ = $$\int {\left( {\left( {1 + x} \right) - {3 \over {1 + x}}} \right)\left( {{1 \over {1 + x}}} \right)dx} $$
<br><br>$\Rightarrow$ ${y \over {1 + x}}$ = $\int {{{{x^2} + 2x + 1 - 3} \over {{{\left( {x + 1} \right)}^2}}}dx}$
<br><br>$\Rightarrow$ ${y \over {1 + x}}$ = x + ${3 \over {1 + x}}$ + C
<br><br>As y(2) = 0 $\Rightarrow$ x = 2, y = 0
<br><br>$\therefore$ 0 = 2 + ${3 \over {1 + 2}}$ + C
<br><br>$\Rightarrow$ C = -3
<br><br>So solution is ${y \over {1 + x}}$ = x + ${3 \over {1 + x}}$ - 3
<br><br>y(3) means x = 3 and find value of y.
<br><br>${y \over {1 + 3}} = 3 + {3 \over {1 + 3}} - 3$
<br><br>$\Rightarrow$ y = 3
About this question
Subject: Mathematics · Chapter: Differential Equations · Topic: Order and Degree
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