Question

Show that the substitution u = y’ leads to a Bernoulli equation. Solve this equation

xy''=y'+x(y')²

Short answer

The solution is y= -ln |c₁-x²|+c₂

Step-by-step solution

Substituting these values;                                              

Let’s substitute u=y’. Then y’’=u’. Substitute these values into given equation

xy''=y' + x(y')²

xu'=u+ xu²

u'= u/x+u²

u^'- (1/x)u=u²

Which is a Bernoulli’s equation with valuse of n=2.

Substitute to reduce equation to a linear one;

We substitute ω = u^1-2=u^-1 or u=ω^-1 to reduce the equation into linear;

du/dx=du/ω. ω /dx= -ω ^-2 dw/dx

-ω^-2 dw/dx:(1/x)ω ^-1= (ω^-1)²

dω /dx+(1/x)ω = -1

Multiplying the equation to get integrating factor;

Multiply the equation with the integrating factor to get

xdω /dx+ω = -X

integrating we get,

∫ d/dx[xω] = ∫ -xdx

xω =(-1/2)x²+c

ω = -(x/2)+(c/x)

Since, ω =u^-1=1/u and u=y' the solution of the given equation will be

1/u==-x/2+c/x

1/y'= - x²-2c/2x

y'=∫ 2x/(c₁-x²)

y= ∫ 2x/(c₁-x²) dx

Hence the final answer is y= - ln|c₁-x²| + c₂