The solution of ey-x dy/dx = y(sinx + cosx)/(1 + ylogy)
Approach Solution - 1
Approach Solution -2
Solution to the Differential Equation
Given the differential equation:
ey-x dy/dx = y(sinx + cosx)/(1 + ylogy)
Rearrange to separate variables:
e^(y-x) dy/dx = y(sinx + cosx)/(1 + ylogy)
Simplify and integrate both sides:
∫ (1 + logy) e^y dy = ∫ (sinx + cosx) e^x dx
Using integration by parts, the solution is:
logy * e^y - ∫ e^y / y dy = e^x (sinx - cosx) + C
Where C is the constant of integration.
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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