If
$8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right)dy = \left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)^{-1}\,\,dx, \,\,\,\,x > 0$
and $
- 3
- 9
- 16
- 80
The Correct Option is A
Solution and Explanation
Let $4+\sqrt{9+\sqrt{x}}=t$
$\Rightarrow \frac{1}{2 \sqrt{9+\sqrt{x}}} \cdot \frac{1}{2 \sqrt{x}} d x=d t $
$\Rightarrow d y=\frac{d t}{2 \sqrt{t}}$
$\Rightarrow 2 d y=\frac{1}{\sqrt{t}} d t$
$2 y=2 \sqrt{t}+c$
$\Rightarrow 2 y=2 \sqrt{4+\sqrt{9+\sqrt{x}}}+c$
$y(0)=\sqrt{7} $
$\Rightarrow c=0 $
$y=\sqrt{4+\sqrt{9+\sqrt{x}}}$
$y(256)=3$
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations