Let x = x(y) be the solution of the differential equation
\(2ye^{\frac{x}{y^2}}dx+(y^2−4xe^{\frac{x}{y^2}})dy=0 \)
such that x(1) = 0. Then, x(e) is equal to
Let x = x(y) be the solution of the differential equation
\(2ye^{\frac{x}{y^2}}dx+(y^2−4xe^{\frac{x}{y^2}})dy=0 \)
such that x(1) = 0. Then, x(e) is equal to
e loge(2)
-e loge(2)
e2 loge(2)
-e2 loge(2)
The Correct Option is D
Solution and Explanation
The correct answer is (D) : -e2 loge(2)
Given differential equation
\(2ye^{\frac{x}{y^2}}dx+(y^2−4xe^{\frac{x}{y^2}})dy=0 ,x(1)=0\)
\(⇒e^{\frac{x}{y^2}}[2ydx−4xdy]=−y^2dy\)
\(⇒e^{\frac{x}{y^2}}[\frac{2ydx−4xdy}{y^4}]=\frac{−1}{y}dy\)
\(⇒2e^{\frac{x}{y^2}}d(\frac{x}{y^2})=\frac{−1}{y}dy\)
\(⇒2e^{\frac{x}{y^2}}=\) −ln y+c…(i)
Now, using x(1) = 0, c = 2
So, for x(e), Put y = e in (i)
\(2e^{\frac{x}{e^2}}=−1+2 \)
\(⇒\frac{x}{e^2}\) =ln\((\frac{1}{2})\) ⇒x(e)= −e2ln2
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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