Question:
The solution of the differential equation
$\frac{dy}{dx}=\frac{yf '\left(x\right)-y^{2}}{f \left(x\right)}$ is
The solution of the differential equation
$\frac{dy}{dx}=\frac{yf '\left(x\right)-y^{2}}{f \left(x\right)}$ is
Updated On: Apr 19, 2024
- $f(x) = y+C$
- $f(x) = y(x+C)$
- $f(x) = x+C$
- None of the above
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The Correct Option is B
Solution and Explanation
The given equation is
$\frac{d y}{d x}=\frac{y f^{\prime}(x)-y^{2}}{f(x)} $
$\Rightarrow y f^{\prime}(x) d x-f(x) d y=y^{2} d x $
$\Rightarrow \frac{y f^{\prime}(x) d x-f(x) d y}{y^{2}}=d x $
$\Rightarrow d\left\{\frac{f(x)}{y}\right\}=d x$
On integration, we get
$ \frac{f(x)}{y}=x+C $
$\Rightarrow f(x)=y(x+C)$
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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