If $x dy = y(dx+y dy),y(1)=1$ and $y(x) > 0$. Then, $y(-3)$ is equal to
- 3
- 2
- 1
- 0
The Correct Option is A
Solution and Explanation
The correct option is(A): 3
Given, \(x dy=y(dx+y dy), y > 0\)
\(\Rightarrow \, \, \, \, \, \, x \, dy-y \, dx=y^2dy\)
\(\Rightarrow \, \, \, \, \, \, \frac {x \, dy-y \, dx }{y^2}=dy \Rightarrow d\bigg ( \frac {x}{y} \bigg )=-dy\)
On integrating both sides, we get
\(\hspace10mm \frac {x}{y}=-y+c \hspace10mm ...(i)\)
Since, \(y(1)=1 \Rightarrow x=1,y=1\)
\(\therefore \hspace8mm c=2\)
Now, E (i) becomes, \(\frac {x}{y}+y = 2\)
Again, for x=-3
\(\Rightarrow \hspace5mm -3+y^2=2y\)
\(\Rightarrow \hspace5mm y^2-2y-3=0\)
\(\Rightarrow \hspace5mm (y+1)(y-3)=0\)
As y>0, take y=3, neglecting y=-1.
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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