objetosc bryly ograniczonej powierzchniami Paulina: Rozwiaz problem poczatkowy y'−9x²y=(x⁵+x²)³√y²

Mariusz: Równanie Bernoulliego y'−9x²y=(x⁵+x²)³√y²

y'
	−9x23√y=(x5+x2)
3√y2

u=³√y

	1
u'=		y−2/3y'
	3

3u'−9x²u=(x⁵+x²)

	1
u'−3x2u=		x2(x3+1)
	3

u'−3x²u=0 u'=3x²u

u'
	=3x2
u

du
	=3x2dx
u

ln|u|=x³+ln|C₁| u=C₁e^x3 u(x)=C₁(x)e^x3

	1
C1'(x)ex3+3x2C1(x)ex3−3x2C1(x)ex3=		x2(x3+1)
	3

	1
C1'(x)ex3=		x2(x3+1)
	3

	1
C1'(x)=		x2(x3+1)e−x3
	3

	1		1
∫(x3+1)x2e−x3dx=−		(x3+1)e−x3+		∫3x2e−x3dx
	9		9

	1		1
∫(x3+1)x2e−x3dx=−		(x3+1)e−x3−		e−x3+C2
	9		9

	1
∫(x3+1)x2e−x3dx=−		(x3+2)e−x3e−x3+C2
	9

	1
C1(x)=−		(x3+2)e−x3e−x3+C2
	9

u(x)=C₁(x)e^x3

	1
u(x)=−		(x3+2)+Cex3
	9

	1
3√y=−		(x3+2)+Cex3
	9

	1
y=(−		(x3+2)+Cex3)3
	9