Euler s differential equation

The differential equations variant coefficients
A second order nonhomogeneous equation with variant coefficients is written as
P(x) y^ +Q(x) y^ +R(x)y=F(x)
I- Euler s differential equation
In this section we want to look for solutions to the differential equation in the form
a_2 x^2 y^ +a_1 xy^ +a_0 y=f(x)
These type of differential equations are called Euler equations.
This differential equation can be transformed in to a second order homogeneous equation with constant coefficients with use of the substitution: x=e^t.
Differentiating with respect to t gives:
dx/dt=e^t so dx/dt=x
In the Euler equation we see the term xy^ which we need to substitute for, so we will multiply both sides of

dy/dx×dx/dt=x dy/dx
This simplifies to
Differentiating this with respect to x gives:
In the Euler equation we see the term xy^ which we need to substitute for, so we will multiply the left side of the above equation by and the right hand side by x (Remember they are equal) gives:
x^2 y^ =(d^2 y)/(dt^2 )-dy/dt
So under this substitution the Euler equation becomes:
?(a_2 (d^2 y)/(dt^2 )+(a_1-a_2 ) dy/dt+a_0 y=f(e^t ) )
This is a second order linear equation with constant coefficients.

Example 1: Solve x^2 y^ +xy^ +y=2
Solution : The given differential equation is Euler equation
Put x=e^t
Then we get (d^2 y)/(dt^2 )+(1-1) dy/dt+y=2
Or (d^2 y)/(dt^2 )+y=2
m^2+1=0 ? m=±i
y_h=c_1 sin?t+c_2 cos?t
Let y_p=A then dy/dt=(d^2 y)/(dt^2 )=0 ? A=2 ? y_p=2
y=y_h+y_p=c_1 sin?t+c_2 cos?t+2
y=y_h+y_p=c_1 sin?t+c_2 cos?t+2
We have x=e^t ? t=ln?x
Then y=c_1 sin?(ln?x )+c_2 cos?(ln?x )+2
Example 2: Solve x^2 y^ -2xy^ +2y=4x^3
Solution : The given differential equation is Euler equation
Put x=e^t
Then we get (d^2 y)/(dt^2 )+(-2-1) dy/dt+2y=4e^3t ? (d^2 y)/(dt^2 )-3 dy/dt+2y=4e^3t
m^2-3m+2=0 ?(m-1)(m-2)=0 ? m_1=1 and m_2=2
y_h=c_1 e^t+c_2 e^2t
Let y_p=Ae^3t then dy/dt=3Ae^3t and (d^2 y)/(dt^2 )=9Ae^3t

9Ae^3t-9Ae^3t+2Ae^3t=4e^3t ? A=2 ? y_p=2e^3t
So y=y_h+y_p=c_1 e^2t+c_2 e^t+2e^3t
We have x=e^t
Then y=c_1 x^2+c_2 x+2x^3

Example 3: Solve x^2 y^ +6xy^ +6y=ln?x
Solution : The given differential equation is Euler equation
Put x=e^t
Then we get (d^2 y)/(dt^2 )+(6-1) dy/dt+6y=t ? (d^2 y)/(dt^2 )+5 dy/dt+6y=t
m^2+5m+6=0?(m+3)(m+2)=0 ? m_1=-3 ,m_2=-2
y_h=c_1 e^(-3t)+c_2 e^(-2t)
Let y_p=At+B then dy/dt=A and (d^2 y)/(dt^2 )=0
5A+6At+6B=t
6A=1 ? A=(1?6)
5A+6B=0 ? B=-(5?36)
y_p=(1?6)t-(5?36)
y=y_h+y_p=c_1 e^(-3t)+c_2 e^(-2t)+(1?6)t-(5?36)
y=c_1 x^(-3)+c_2 x^(-2)+(1?6) ln?x-(5?36)


Solve the differential equations
(1) x^2 y^ -5xy^ +8y=2x^2
(2) x^2 y^ -xy^ -3y=x^5
(3) x^2 y^ +5xy^ +4y=1/x^2
(4) x^2 y^ +3xy^ +5y=ln?x