# How Do I Solve a Second Order ODE with Non-Constant Coefficients?

• Remixex
In summary, the differential equation was created by the speaker and they have never solved a second order ODE with functions as coefficients. They usually solve with constant coefficients. The only way to solve these equations is via Cauchy-Euler methods or Riccati equation. However, the speaker created an equation that can't be solved the way they want to.
Remixex

## Homework Statement

OK, this differential equation was technically created by me, because i need to clear my doubts.
Y'' + sqrt(X)*Y' + X^3*Y=3sin(x)
and actually just any initial conditions as long as the solution is something i can understand, let me expand my doubt further.

I've never solved a second order ODE with functions as coefficients, I've always done it with constant coefficients (because those are the ones that describe oscillations), i usually solve with variation of parameters, Y=Yh+Yp where Yh=C1*Y1(x) +C2 Y2(x) calculated with the "D" operator, and then Yp=U1*Y1 + U2*Y2 and U1 and U2 are calculated integrating the division of the respective wronskians.
The only ways i have to solve these "variable coefficients" equations are via Cauchy-Euler methods (X^m) or Riccati equation knowing one solution.
I actually had to be told this is solved via variation of parameters, and i have no idea how.

## Homework Equations

Y'' +q(x)Y' + p(x) Y = g(x)
(In fact if you have any other example stored in that clarifies my doubt i'd greatly appreciate it, as long as it fits this equation above)

## The Attempt at a Solution

The differential operator would be D^2 + sqrt(x)*D +X^3=0, but really i don't know how to solve for "D" (to get the general expression for Y1 and Y2) if i also have to solve for X
Thanks in advance, this might actually be a very basic doubt (it has been in my mind a long while, i know how to solve some PDEs and membrane problems with different boundary conditions but not this)

I would look for power series solution to that.

Oh snap, it seems i just created an equation that can't be solved teh way i want to x.x, thanks for your help, i'll go with my doubt directly to the teacher

## 1. How do I know which method to use to solve a differential equation?

There are several methods for solving differential equations, including separation of variables, substitution, and integrating factors. The best method to use will depend on the type of differential equation and its initial conditions. It is important to familiarize yourself with each method and practice solving different types of equations to determine the most appropriate method.

## 2. What are the steps to solving a differential equation?

The general steps for solving a differential equation are as follows: 1) Identify the type of differential equation (e.g. first-order, second-order, etc.) 2) Check for any special conditions or terms (e.g. initial conditions, homogeneous equation) 3) Use an appropriate method to solve the equation 4) Check the solution for accuracy by plugging it back into the original equation.

## 3. Can differential equations be solved numerically or do they require an analytical solution?

Differential equations can be solved both numerically and analytically. Numerical solutions use algorithms and computer programs to approximate the solution, while analytical solutions use mathematical techniques to find an exact solution. In some cases, a numerical approach may be necessary if an analytical solution cannot be found.

## 4. What are some real-world applications of differential equations?

Differential equations are used to model and describe a wide range of phenomena in various fields, including physics, engineering, biology, economics, and more. Examples include population growth, heat transfer, motion of objects, chemical reactions, and electrical circuits.

## 5. How can I check if my solution to a differential equation is correct?

The best way to check the accuracy of a solution to a differential equation is to plug it back into the original equation and see if it satisfies the equation. If the solution satisfies the equation, then it is most likely correct. Additionally, you can compare your solution to a known solution or use a graphing tool to visualize the solution and compare it to the original equation.

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