dooogle said:Homework Statement
find general solution of
dy/dx=xy+x
using integrating factor
Homework Equations
The Attempt at a Solution
rearange
dy/dx-xy=x
take integrating factor as
e^int(-x)=e^-(x^2)/2
multiply throughout
dy/dx(e^-(x^2)/2)y=(e^-(x^2)/2)x
integrate
e^-(x^2)/2*y=int (e^-(x^2)/2)*x dx
dont know how to integrate (e^-(x^2)/2)*x
It's the other way around, plus you left off the factor of x in the integral on the right.dooogle said:i let u=-(x^2)/2
du/dx=-2x/2=-x
du=-x dx
so -int e^u du = int e^-(x^2)/2 dx
In the line above, the arbitrary constant comes in as soon as you have the antiderivative. Also, your 2nd exponential expression is not quite right.dooogle said:=-e^u =-e^-(x^2)/2 +c
The purpose of integrating e^-(x^2)/2 is to find the area under the curve of the function. This can be useful in various fields such as physics and statistics, where the area represents a probability or a physical quantity.
It is difficult to integrate e^-(x^2)/2 because it is a non-elementary function, meaning it cannot be expressed in terms of elementary functions such as polynomials, exponential, trigonometric, or logarithmic functions. As a result, it requires more advanced techniques to be integrated.
The most common methods used to integrate e^-(x^2)/2 are substitution, integration by parts, and special functions such as the error function. These techniques involve manipulating the function algebraically or using specific properties to simplify the integration process.
No, e^-(x^2)/2 cannot be integrated exactly. However, it can be approximated using numerical integration methods such as the trapezoidal rule or Simpson's rule. These methods divide the area under the curve into smaller trapezoids or parabolas, respectively, and calculate the sum of their areas to approximate the overall area.
Integrating e^-(x^2)/2 has various real-world applications, including calculating the probability of events in statistics, determining the concentration of molecules in chemistry, and finding the displacement of an object given its velocity in physics. It is also used in image processing and signal analysis to filter out noise and enhance signals.