prosze o pomoc w rozniczkach Asiaaaa: 1. y^'/e^x= 1/e^y 2. y^'=√2x+y−2 3.y^'+¹_x y=8 4. y^'+6y=x²+1

Asiaaaa: bardzo prosze o pomoc

Mariusz: 1. y'e^−x=e^−y e^ydy=e^xdx e^y=e^x+C y=ln(|e^x+C|) 2. u=2x+y u'=2+y' u'−2=√u−2

du
	=dx
√u

2√u=x+C 4u=(x+C)²

	x+C
u=(		)2
	2

	x+C
2x+y=(		)2
	2

	x+C
y=(		)2−2x
	2

3.

	y
y'+		=8
	x

	y
y'+		=0
	x

	y
y'=−
	x

dy		dx
	=−
y		x

ln|y|=−ln|x|+C

	1
y=C
	x

	1
y(x)=C(x)
	x

	1		1		1
C'(x)		−C(x)		+C(x)		=8
	x		x2		x2

	1
C'(x)		=8
	x

C'(x)=8x C(x)=4x²+C₁

	4x2+C1
y=
	x

	C1
y=4x+
	x

4. y'+6y=x²+1 y'+6y=0 y'=−6y

dy
	=−6dx
y

ln|y|=−6x+C y=Ce^−6x y(x)=C(x)e^−6x C'(x)e^−6x−6C(x)e^−6x+6C(x)e^−6x=x²+1 C'(x)e^−6x=x²+1 C'(x)=(x²+1)e^6x

	1		1
C(x)=		(x2+1)e6x−		∫xe6xdx
	6		3

	1		1		1		1
C(x)=		(x2+1)e6x−		(		xe6x−		∫e6xdx)
	6		3		6		6

	1		1		1		1
C(x)=		(x2+1)e6x−		(		xe6x−		e6x)+C1
	6		3		6		36

	1		1		1
C(x)=		(x2+1)e6x−		xe6x+		e6x+C1
	6		18		108

	1
C(x)=		(18x2−6x+19)e6x+C1
	108

	1
y=		(18x2−6x+19)+C1e−6x
	108

Asiaaaa: dziekuje bardzo