Question:
If the general solution of the differential equation $\left(\text{y}\right)^{'}=\frac{\text{y}}{\text{x}}+\phi\left(\frac{\text{x}}{\text{y}}\right)$ , for some function $\phi$ , is given by $yln\left|\right.\text{c}\text{x}\left|\right.=\text{x}$ , where $\text{c}$ is an arbitrary constant, then $\phi\left(2\right)$ is equal to $\left(\right)\text{here} , \, \left(\text{y}\right)^{'}=\frac{d y}{d x}\left(\text{y}\right)^{'}=\frac{d y}{d x}$
If the general solution of the differential equation $\left(\text{y}\right)^{'}=\frac{\text{y}}{\text{x}}+\phi\left(\frac{\text{x}}{\text{y}}\right)$ , for some function $\phi$ , is given by $yln\left|\right.\text{c}\text{x}\left|\right.=\text{x}$ , where $\text{c}$ is an arbitrary constant, then $\phi\left(2\right)$ is equal to $\left(\right)\text{here} , \, \left(\text{y}\right)^{'}=\frac{d y}{d x}\left(\text{y}\right)^{'}=\frac{d y}{d x}$
Updated On: Jul 28, 2022
- $-4$
- $- \frac{1}{4}$
- $\frac{1}{4}$
- $4$
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The Correct Option is B
Solution and Explanation
$\text{given} \text{:} \left(\text{y}\right)^{'} = \frac{\text{y}}{\text{x}} + \phi \left(\frac{\text{x}}{\text{y}}\right)$
As $\text{y ln} \left(\text{cx}\right) = \text{x} \, ? \, \left(\text{y}\right)^{'} \text{ln} \left(\text{cx}\right) + \text{y} \frac{1}{\text{cx}} \text{c} = 1$
$? \, \left(\text{y}\right)^{'} \left(\frac{\text{x}}{\text{y}}\right) + \frac{\text{y}}{\text{x}} = 1$
$? \, \, \, \frac{1 - \left(\frac{\text{y}}{\text{x}}\right)}{\left(\frac{\text{x}}{\text{y}}\right)} = \left(\frac{\text{y}}{\text{x}}\right) + \phi \left(\frac{\text{x}}{\text{y}}\right)$
Put $\frac{x}{y} = 2 \, ? \, \frac{1 - \left(\frac{1}{2}\right)}{\left(\frac{2}{1}\right)} = \left(\frac{1}{2}\right) + \phi \left(\frac{2}{1}\right)$
$? \, \, \, \phi \left(2\right) = - \frac{1}{4}$
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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