Question

Discuss how to find an alternative two-parameter family of solutions for the nonlinear differential equation y’’ = 2x( y’)2 in Example 1.

Short answer

The solution of the above question is y= 1/2c₁ln x+c₁/x-c₁+c₂.

Step-by-step solution

Solving of second order and non-linear differential equation:

Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions. To decompose a fraction, you first factor the denominator.

Method of reduction order will be applied:

In this question we will apply the conceptMethod of reduction order;

y^''=2x (y^')²

y^''=dv/dx

dv/dx=2xv²

∫ 1/v²dv=2xv²

∫1/v² dv = 2∫ xdx

∫ v^-2dv=2∫ xdx

-v^-1=2× 1/2 x² c₁²

-1/v= x²+ c₁²

v= -1/x²+c₁²

dy/dx=-1/x²-c₁²

Partial Differentiation will be applied;

Using Partial fraction decomposition, we will expand the answer:

-1/x²-c₁²= - 1/(x-c₁)(x+c₁)

= (A+B)x + (Ac₁-Bc₁)/ (x-c₁) (x+c₁)

A+B=0

Ac₁-Bc₁= -1

A= -1/2c₁, B=1/2c₁

dy/dx=-1/2c₁ 1/(x-c₁) +1/2c₁ 1/x+c₁

∫ dy= ∫ [(-1/2c₁) (1/x-c₁)]+ [(1/2c₁)(1/x+c₁)] dx

y= 1/2c₁ ln (x+c₁/x-c₁) +c₂

Hence, the final answer isy= 1/2c₁ ln (x+c₁/x-c₁) +c₂.