Find a general solution of each of the following differential

Kelly Nelson

Kelly Nelson

Answered question

2021-12-28

Find a general solution of each of the following differential equations: dydx=x6y42yx

Answer & Explanation

macalpinee3

macalpinee3

Beginner2021-12-29Added 29 answers

dydx+2xy=x5y4
1y4dydx+2x1y3=x5
Let, 1y3=t3y4dydx=dtdx
1dx+2xt=x5
dtdx6xt=3x5
I.F=e6xdx=e6lnx=eln(1x6)=1x6
t(1x6)=3x51x6dx+c
=31xdx+c=3ln(x)+c
t=x6(3ln(x)+c)
1y3=t
1y3=x6(3ln(x)+c)
y(x)=13{x6(3ln(x)+c)}
or y(x)=3{1}3{x6(3ln(x)+c)}
or
Mary Herrera

Mary Herrera

Beginner2021-12-30Added 37 answers

dydx=x6y42yx
yx=y(x6y4y2yy)x
yx=y(x6y412)x
yx=y(x6y32)x
y=y(x6y32)
y=yx6y3y2
y=yy3x62y
y=y1+3x62y
y=y4x62y
y+(y4x6+2y)=(y4x62y)+(y4x6+2y)
yy4x6+2y=y4x62yy4x6+2y
y+2yy4x6=y4x6y4x62y+2y
3yy4x6=0
y(3yyy4x6y)=0
y(3(y41x6))=0
y(3(y3x6))=0
y(y3x6+3)=0
y=0
y3x6+3=0
karton

karton

Expert2022-01-09Added 613 answers

Rewrite in form of a first order Bernoulli ODE
y+2xy=x5y4
The general solution is obtained by substituting v=y1n and solving 11nv+p(x)v=q(x)
Transform to 11nv+p(x)v=q(x):13v+2vx=x5
Solve 13v+2vx=x5:v=3ln(x)x6+c1x6
Substitute back v=y3:y3=3ln(x)x6+c1x6
Isolate y: y=1x6(3ln(x)+c1)3
y=x6(3ln(x)+c1)3

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