MATH 302-010
Ordinary Differential Equations
Prof. D. A. Edwards
Copyright 2024

Euler Equations (Revised)

We begin with our standard command:

Quit[]

Euler Equations

Mathematica can solve Euler equations directly, with or without boundary conditions:

rert=x^2*y''[x]-3*x*y'[x]-y[x]==0
DSolve[rert,y[x],x]
DSolve[{rert,y[1]==1,y'[1]==0},y[x],x]

Note that Mathematica doesn’t automatically insert the absolute value of x.  In particular, this causes problems if you have initial conditions for x<0.  It is beyond the scope of what I want to cover to describe a workaround, so no Mathematica-based questions will have such initial conditions.

Here’s an equation with a complex root:

comrt = 2*x^2*y''[x]-3*x*y'[x]+5*y[x]==0
DSolve[comrt,y[x],x]
DSolve[{comrt,y[1]==1,y'[1]==0},y[x],x]

Note that Mathematica does convert to sine and cosine.

Here’s an equation with a double root of λ=1:

dbrt = x^2*y''[x]+3*x*y'[x]+y[x]==0
DSolve[dbrt,y[x],x]
DSolve[{dbrt,y[1]==1,y'[1]==0},y[x],x]

Exercise

1. Write the general solution of 7(x^3)(((d^3)y)/(dx^3))+3(x^2)(((d^2)y)/(dx^2))+x(dy/dx)+9y=0.

Solution

eulprob = 7*x^3*D[y[x],{x,3}]+3*x^2*y''[x]+x*y'[x]+9*y[x]==0
DSolve[eulprob,y[x],x]