Trying to solve a second order ODE

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SUMMARY

The discussion focuses on solving the second order ordinary differential equation (ODE) for the function y(x) to demonstrate that the solution is y(x) = sin(x)/x. The user employs the substitution y(x) = h(x)/x, leading to the derived equation h''(x) = -x. The general solution for h(x) is expressed as h(x) = A sin(x) + B cos(x). The user encounters confusion regarding the initial condition y(0) = 1, questioning its validity due to the singularity of y at x = 0.

PREREQUISITES
  • Understanding of second order ordinary differential equations (ODEs)
  • Familiarity with the method of substitution in differential equations
  • Knowledge of limits and continuity, particularly lim x->0 sin(x)/x
  • Basic concepts of initial conditions in differential equations
NEXT STEPS
  • Review the method of solving second order ODEs with initial conditions
  • Study the implications of singular points in differential equations
  • Learn about the behavior of functions near singularities, specifically sin(x)/x
  • Explore the application of initial conditions in the context of differential equations
USEFUL FOR

Students studying differential equations, mathematicians interested in ODE solutions, and educators teaching calculus and analysis concepts.

cahill8
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Homework Statement



I'm trying to solve a second order ODE for y(x) to show that the solution is y(x)=sin(x)/x. We've been told to use the substitution y(x)=h(x)/x. I've got to the stage of solving for h(x), arriving at h''(x)=-x. Using the general solution, h(x)=A sin(x) + B cos(x) and substiting this into the original equation for y(x) I get y(x)=(A sin(x) + B cos(x))/x

So all that's left to do it seems is to use the inital conditions to show A=1 and B=0 however the problem says to use the inital condition y(0)=1. This doesn't make sense to me since y is singular at x=0. Is this inital condition a mistake or am I missing something?


Homework Equations


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The Attempt at a Solution


given above
 
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hi cahill8! :smile:

limx->0 sinx/x = 1 :wink:
 

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