Question:
If $y'' - 3y' + 2y = 0$ where $y(0) = 1$, $y'(0) = 0$, then the value of $y$ at $x \,= log_e \,2$ is
If $y'' - 3y' + 2y = 0$ where $y(0) = 1$, $y'(0) = 0$, then the value of $y$ at $x \,= log_e \,2$ is
Updated On: Feb 15, 2024
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The Correct Option is D
Solution and Explanation
$\frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+2 y=0$
The corresponding equation is $m^{2}-3 m+2=0$
$\therefore$ General solution of given equation
$y=A e^{x}+B e^{2 x}$
$y^{'}=A e^{x}+2 B e^{2 x}$
At $x=0, y=1 $
$\Rightarrow A+B=1$
and $x=0, y^{'}=0$
$ \Rightarrow A+2 B=0$
Solving these equation $A=2,\, B=1$
$\therefore y=2 e^{x}-e^{2 x}$
At $x=\log 2, \,y=0$
The corresponding equation is $m^{2}-3 m+2=0$
$\therefore$ General solution of given equation
$y=A e^{x}+B e^{2 x}$
$y^{'}=A e^{x}+2 B e^{2 x}$
At $x=0, y=1 $
$\Rightarrow A+B=1$
and $x=0, y^{'}=0$
$ \Rightarrow A+2 B=0$
Solving these equation $A=2,\, B=1$
$\therefore y=2 e^{x}-e^{2 x}$
At $x=\log 2, \,y=0$
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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