Variation of parameters

II-Variation of parameters
The idea of variation of parameters is look for functions such that satisfies where are solutions of .We will arbitrarily decide only for which also satisfy
If we substitution into the differential equation, we get

Then we solve and to find and then we get and
Example 1:
Solution:




II-Variation of parameters
The idea of variation of parameters is look for functions v_1 (x) and v_2 (x) such that y_p=v_1 y_1+v_2 y_2 satisfies ay^ +by^ +cy=R(x) where y_1 and y_2 are solutions of ay^ +by^ +cy=0.We will arbitrarily decide only for v_1 and v_2 which also satisfy v_1^ y_1+v_2^ y_2=0 ?(1)
If we substitution y_p=v_1 y_1+v_2 y_2 into the differential equation, we get
v_1^ y_1^ +v_2^ y_2^ =R(x) ?(2)
Then we solve (1) and (2) to find v_1^ and v_2^ then we get v_1 and v_2.
Example 1: Solve y^ +y=csc?x
Solution:
m^2+1=0 ?m=?i
y_h=c_1 cos?x+c_2 sin?x
v_1^ (x) cos?x+v_2^ (x) sin?x=0
-v_1^ (x) sin?x+v_2^ (x) cos?x=csc?x
v_1^ (x)=|?(0&sin?x@csc?x&cos?x )|/|?(cos?x&sin?x@?-sin??x&cos?x )| =(-csc?x sin?x)/(cos^2?x+sin^2?x )=-1
? v_1 (x)=-x
v_2^ (x)=|?(cos?x&0@?-sin??x&csc?x )|/|?(cos?x&sin?x@?-sin??x&cos?x )| =(cos?x csc?x)/(cos^2?x+sin^2?x )=cot?x
? v_2 (x)=ln?|sin?x |
y_p=v_1 y_1+v_2 y_2=-x cos?x+sin?x ln?|sin?x |
y=y_h+y_p=c_1 cos?x+c_2 sin?x-x cos?x+sin?x ln?|sin?x |

Example 2: Solve y^ -5y^ +6y=x^2 e^3x
Solution
m^2-5m+6=0 ?(m-3)(m-2)=0 ? m_1=3 and m_2=2
y_h=c_1 e^3x+c_2 e^2x
v_1^ (x) e^3x+v_2^ (x) e^2x=0
3v_1^ (x) e^3x+2v_2^ (x) e^2x=x^2 e^3x
v_1^ (x)=|?(0&e^2x@x^2 e^3x&2e^2x )|/|?(e^3x&e^2x@3e^3x&2e^2x )| =(-x^2 e^3x×e^2x)/(e^3x×2e^2x-3e^3x×e^2x )=x^2
? v_1 (x)=x^3?3
v_2^ (x)=|?(e^3x&0@3e^3x&x^2 e^3x )|/|?(e^3x&e^2x@3e^3x&2e^2x )| =(e^3x×x^2 e^3x)/(e^3x×2e^2x-3e^3x×e^2x )=(x^2 e^6x)/(-e^5x )=-x^2 e^x

v_2 (x)=-x^2 e^x+2xe^x-2e^x=e^x (-x^2+2x-2)
y_p=v_1 y_1+v_2 y_2=(x^3?3) e^3x+e^x (-x^2+2x-2) e^2x=e^3x ((x^3?3)-x^2+2x-2)
y=y_h+y_p=c_1 e^3x+c_2 e^2x+e^3x ((x^3?3)-x^2+2x-2)

Solve the differential equations
(1) y^ +9y=tan?3x (2) y^ +4y=sin?2x sec^2?2x
(3) y^ +2y^ +2y=3e^x sec?x (4) y^ -2y^ +y=14x^(3?2) e^x









Example 2:
Solution












Solve the differential equations