Question:

If the solution of the differential equation
\(\frac{dy}{dx}\) \(+ e^x(x² - 2)y = (x^2 - 2x)(x^2 - 2)e^{2x}\)
satisfies y(0) = 0, then the value of y(2) is ______.

Updated On: Mar 20, 2025
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The Correct Option is C

Solution and Explanation

The correct answer is (C) : 0
\(\frac{dy}{dx} + e^x(x^2-2)y=(x^2-2x)(x^2-2)e^{2x }\)
Here, I.F.
\(=\) \(e^{\int{e^x(x² - 2)dx}}\)
\(=\) \(e^{(x² - 2x)e^x}\)
∴ Solution of the differential equation is 
\(y.e^{(x² - 2x)e^x} = \int{(x² - 2x)(x² - 2)e^{2x}.e^{(x² - 2x)e^x }dx}\)
\(= \int{ (x² - 2x)e^x.(x² - 2)e^x.e(x² - 2x)e^x dx}\)
Let
\((x² - 2x)e^x = t\)
\(∴ (x² - 2)e^x dx = dt\)
\(y.e(x² - 2x)e^x = ∫ t.e^tdt\)
\(y.e(x² - 2x)e^x = (x² - 2x - 1)e^{(x² - 2x)e^x} + c\)
\(∴ y(0) = 0\)
\(∴ c = 1\)
\(∴ y = (x² - 2x - 1) + e(2x - x²)e^x\)
\(∴ y(2) = -1 + 1 = 0\)

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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