Let the solution curve of the differential equation \(x\frac{dy}{dx}-y=\sqrt{y^2+16x^2},\) y(1) = 3 be y = y(x). Then y(2) is equal to
- 15
- 11
- 13
- 17
The Correct Option is A
Solution and Explanation
The correct option is(A): 15
\(x\frac{dy}{dx}-y=\sqrt{y^2+16x^2}\)
y = 4x tan θ
log |secθ + tanθ| = log |x| + C
y(1) = 3 ⇒ 3 = 4 tanθ
⇒ C = ln 2
∴ |secθ + tanθ| = 2|x|
To find y(2) put x = 2
⇒tanθ= \(\frac{y}{8}\)
(secθ + tanθ)2 = 16
⇒ y = 15
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Concepts Used:
Types of Differential Equations
There are various types of Differential Equation, such as:
Ordinary Differential Equations:
Ordinary Differential Equations is an equation that indicates the relation of having one independent variable x, and one dependent variable y, along with some of its other derivatives.
\(F(\frac{dy}{dt},y,t) = 0\)
Partial Differential Equations:
A partial differential equation is a type, in which the equation carries many unknown variables with their partial derivatives.
Linear Differential Equations:
It is the linear polynomial equation in which derivatives of different variables exist. Linear Partial Differential Equation derivatives are partial and function is dependent on the variable.
Homogeneous Differential Equations:
When the degree of f(x,y) and g(x,y) is the same, it is known to be a homogeneous differential equation.
\(\frac{dy}{dx} = \frac{a_1x + b_1y + c_1}{a_2x + b_2y + c_2}\)
Read More: Differential Equations