Euler equation A second-order Euler equation is one of the form ax^2y\" + bxy\'

Question

Euler equation A second-order Euler equation is one of the form ax^2y" + bxy' + cy = 0, where a, b, and c are constants. (a) Show that if x > 0, then the substitution u = ln x transforms Eq(1) into the constant-coefficient, linear differential equation a d^2y/du^2 + (b - a)dy/du + cy = 0, with independent variable u. We have learned techniques to solve such constant-coefficient, linear, homogeneous differential equation. (b) Write the characteristic equation corresponding to (2). (c) What condition on a b, and c guarantees that the roots of the characteristic equation are real and distinct? (d) Find a general solution for x > 0 of the Euler equation 4x^2y" + 8xy' - 15y = 0.

Explanation / Answer

a)

A second order euler equation is given by :
ax^2*d^2y/dx^2 + bx*dy/dx + cy = 0
where a, b, & c are constants.

We are given for x > 0, substitution u = ln x, means
x = e^u => dx/du = e^u

Now since y is function of u through x, so by chain rule :
dy/du = dy/dx * dx/du
= dy/dx * e^u
= dy/dx * x

Now,
d^2y/du^2 = d/du (x*dy/dx)
= dx/du*dy/dx + x d/du (dy/dx)
= e^u * dy/dx + x*d^2y/dx^2*dx/du
= x*dy/dx + x^2*d^2y/dx^2
= dy/du + x^2*d^2y/dx^2
or, x^2*d^2y/dx^2 = d^2y/du^2 – dy/du

Now putting these values in the given second order euler equation, we get

a*( d^2y/du^2 – dy/du) + bx*(1/x)*dy/du + cy = 0

a*d^2y/du^2 + (b – a)*dy/du + cy = 0

Proved

b)

Characteristic equation of above transformed D.E is:

am^2 + (b – a)m + c = 0

where, m represents roots of the DE, since it's a quadratic equation so roots will be m1, m2.

d)

Now general solution of :
4x^2*d^2y/dx^2 + 8x*dy/dx - 15y = 0
Its transformed can be written in the form we have derived in part-a like :

4d^2y/du^2 + (8 - 4)*dy/du - 15y = 0

Characteristic equation of this is :

4m^2 + 4m - 15 = 0
Upon solving this quadratic equation we get

4m^2 + 10m - 6m - 15 = 0
(2m + 5)(2m – 3) = 0
m = -5/2 & 3/2

So,
Fundamental set of solutions = {e^-5t/2, e^3t/2}

General solution is : y(t) = Ae^-5t/2 + Be^3t/2 (here A & B are real valued constants)

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