Question

solve x²y''-4xy'+6y=x²+log x​

Answer 1

222+4+6=ln(2)=2ln()
x
2
d
2
y
d
x
2
+
4
x
d
y
d
x
+
6
y
=
ln
⁡
(
x
2
)
=
2
ln
⁡
(
x
)


To solve this equation,we take …………….

=ln()⇒=1⇒=
z
=
ln
⁡
(
x
)
⇒
d
y
d
x
=
1
x
d
y
d
z
⇒
x
d
y
d
x
=
d
y
d
z


Similarly, 222=22+
x
2
d
2
y
d
x
2
=
d
2
y
d
z
2
+
d
y
d
z


Putting this in original equation,then we get……..

22++4+6=2⇒22+5+6=2
d
2
y
d
z
2
+
d
y
d
z
+
4
d
y
d
z
+
6
y
=
2
z
⇒
d
2
y
d
z
2
+
5
d
y
d
z
+
6
y
=
2
z


Now it's corresponding homogeneous equation is…..

22+5+6=0
d
2
y
d
z
2
+
5
d
y
d
z
+
6
y
=
0


The auxiliary equation is…….

2+5+6=0⇒(+3)(+2)=0⇒=−3,−2
m
2
+
5
m
+
6
=
0
⇒
(
m
+
3
)
(
m
+
2
)
=
0
⇒
m
=
−
3
,
−
2


Complimentary function is…….

=−3+−2=−3+−2
y
c
=
A
e
−
3
z
+
B
e
−
2
z
=
A
x
−
3
+
B
x
−
2