general solution of (4y+x^3)dx+xdy=0
Answers
x
dx
dy
+4y=x
3
e
x
Dividing both sides by xx ,
\dfrac{dy}{dx}+\dfrac4xy=x^2e^x
dx
dy
+
x
4
y=x
2
e
x
...(i)
On comparing this with \dfrac{dy}{dx}+Py=Q
dx
dy
+Py=Q , we get
P=\dfrac4x,\ Q=x^2e^xP=
x
4
, Q=x
2
e
x
Now, integrating factor, I.F.=e^{\int Pdx}I.F.=e
∫Pdx
I.F.=e^{\int \frac4xdx}=e^{4\ln x}=e^{\ln x^4}=x^4I.F.=e
∫
x
4
dx
=e
4lnx
=e
lnx
4
=x
4
On multiplying (i) by this I.F., we get,
I.F.\times y=\int Q\times I.F. \ dxI.F.×y=∫Q×I.F. dx
\Rightarrow x^4\times y=\int x^2e^x\times x^4\ dx⇒x
4
×y=∫x
2
e
x
×x
4
dx
\Rightarrow yx^4=\int x^6e^x\ dx⇒yx
4
=∫x
6
e
x
dx
Now we apply integration by parts on right side, taking x^6x
6
as first function and e^xe
x
as second. Also we need to perform this rule various times further.
\Rightarrow yx^4=x^6e^x-\int 6x^5e^x\ dx \\ \Rightarrow yx^4=x^6e^x-6[x^5e^x-\int 5x^4e^x]\ dx \\ \Rightarrow yx^4=x^6e^x-6x^5e^x+30[x^4e^x-\int 4x^3e^x]\ dx⇒yx
4
=x
6
e
x
−∫6x
5
e
x
dx
⇒yx
4
=x
6
e
x
−6[x
5
e
x
−∫5x
4
e
x
] dx
⇒yx
4
=x
6
e
x
−6x
5
e
x
+30[x
4
e
x
−∫4x
3
e
x
] dx
\\ \Rightarrow yx^4=x^6e^x-6x^5e^x+30x^4e^x-120[x^3e^x-\int 3x^2e^x]\ dx \\ \Rightarrow yx^4=x^6e^x-6x^5e^x+30x^4e^x-120x^3e^x+360[x^2e^x-\int 2xe^x]\ dx
⇒yx
4
=x
6
e
x
−6x
5
e
x
+30x
4
e
x
−120[x
3
e
x
−∫3x
2
e
x
] dx
⇒yx
4
=x
6
e
x
−6x
5
e
x
+30x
4
e
x
−120x
3
e
x
+360[x
2
e
x
−∫2xe
x
] dx
\\ \Rightarrow yx^4=x^6e^x-6x^5e^x+30x^4e^x-120x^3e^x+360x^2e^x-720(xe^x- e^x)+C \\ \Rightarrow yx^4=x^6e^x-6x^5e^x+30x^4e^x-120x^3e^x+360x^2e^x-720xe^x+720 e^x+C
⇒yx
4
=x
6
e
x
−6x
5
e
x
+30x
4
e
x
−120x
3
e
x
+360x
2
e
x
−720(xe
x
−e
x
)+C
⇒yx
4
=x
6
e
x
−6x
5
e
x
+30x
4
e
x
−120x
3
e
x
+360x
2
e
x
−720xe
x
+720e
x
+C
\Rightarrow y=\dfrac{x^6e^x-6x^5e^x+30x^4e^x-120x^3e^x+360x^2e^x-720xe^x+720 e^x+C}{x^4}⇒y=
x
4
x
6
e
x
−6x
5
e
x
+30x
4
e
x
−120x
3
e
x
+360x
2
e
x
−720xe
x
+720e
x
+C
\Rightarrow y=x^2e^x-6xe^x+30e^x-\dfrac{120e^x}{x}+\dfrac{360e^x}{x^2}-\dfrac{720e^x}{x^3}+\dfrac{720 e^x}{x^4}+\dfrac{C}{x^4}⇒y=x
2
e
x
−6xe
x
+30e
x
−
x
120e
x
+
x
2
360e
x
−
x
3
720e
x
+
x
4
720e
x
+
x
4
C