Integration by parts leaves you with an integral that is (up to some constants) the error function.AAO said:Homework Statement
Solve the ODE: y''+x*y'-y=0
Homework Equations
The Attempt at a Solution
Since this is a variable coefficient ODE, I have used the method of reduction of order, and assumed the solution in the form: y=c1*y1+c2*y2
In this case: y1=x, and I have the reached the integral below for the second solution (y2), Can anyone tell me how to approach this integral:
v=integral[ (x^-2) * exp (-0.5*x^2) dx]
The integral of exponential over polynomial is a mathematical expression that represents the area under the curve of an exponential function divided by a polynomial function. It can be written as ∫e^x / (ax^n + bx^(n-1) + ... + cx + d) dx.
Yes, it is possible to solve the integral of exponential over polynomial analytically by using integration techniques such as substitution, integration by parts, or partial fractions.
The integral of exponential over polynomial is commonly used in physics and engineering to solve problems involving exponential growth or decay combined with polynomial functions. It is also used in statistics to calculate the expected value of certain probability distributions.
Yes, the integral of exponential over polynomial can only be solved when the degree of the polynomial is greater than the power of the exponential function. Additionally, the polynomial must not have repeated roots or a zero degree term.
Yes, when the integral of exponential over polynomial cannot be solved analytically, numerical methods such as the trapezoidal rule or Simpson's rule can be used to approximate the value of the integral.