+0  
 
0
364
1
avatar

y'' + 3y = 5lnx help?

Guest Nov 25, 2015

Best Answer 

 #1
avatar
+15

y'' + 3y = 5lnx help?

This is a second-order linear ordinary differential equation. The solution to it is quite involved.

 

Solve ( d^2 y(x))/( dx^2)+3 y(x) = 5 log(x):
The general solution will be the sum of the complementary solution and particular solution.
Find the complementary solution by solving ( d^2 y(x))/( dx^2)+3 y(x)  =  0:
Assume a solution will be proportional to e^(lambda x) for some constant lambda.
Substitute y(x)  =  e^(lambda x) into the differential equation:
( d^2 )/( dx^2)(e^(lambda x))+3 e^(lambda x)  =  0
Substitute ( d^2 )/( dx^2)(e^(lambda x))  =  lambda^2 e^(lambda x):
lambda^2 e^(lambda x)+3 e^(lambda x)  =  0
Factor out e^(lambda x):
(lambda^2+3) e^(lambda x)  =  0
Since e^(lambda x) !=0 for any finite lambda, the zeros must come from the polynomial:
lambda^2+3  =  0
Solve for lambda:
lambda = i sqrt(3) or lambda = -i sqrt(3)
The roots lambda  =  ± i sqrt(3) give y_1(x) = c_1 e^(i sqrt(3) x), y_2(x) = c_2 e^(-i sqrt(3) x) as solutions, where c_1 and c_2 are arbitrary constants.
The general solution is the sum of the above solutions:
y(x)  =  y_1(x)+y_2(x)  =  c_1 e^(i sqrt(3) x)+c_2/e^(i sqrt(3) x)
Apply Euler's identity e^(alpha+i beta) = e^alpha cos(beta)+i e^alpha sin(beta):
y(x)  =  c_1 (cos(sqrt(3) x)+i sin(sqrt(3) x))+c_2 (cos(sqrt(3) x)-i sin(sqrt(3) x))
Regroup terms:
y(x)  =  (c_1+c_2) cos(sqrt(3) x)+i (c_1-c_2) sin(sqrt(3) x)
Redefine c_1+c_2 as c_1 and i (c_1-c_2) as c_2, since these are arbitrary constants:
y(x)  =  c_1 cos(sqrt(3) x)+c_2 sin(sqrt(3) x)
Determine the particular solution to ( d^2 y(x))/( dx^2)+3 y(x)  =  5 log(x) by variation of parameters:
List the basis solutions in y_c(x):
y_(b_1)(x) = cos(sqrt(3) x) and y_(b_2)(x) = sin(sqrt(3) x)
Compute the Wronskian of y_(b_1)(x) and y_(b_2)(x):
(script capital w)(x)  =  |cos(sqrt(3) x) | sin(sqrt(3) x)
( d)/( dx)(cos(sqrt(3) x)) | ( d)/( dx)(sin(sqrt(3) x))|  =  |cos(sqrt(3) x) | sin(sqrt(3) x)
-(sqrt(3) sin(sqrt(3) x)) | sqrt(3) cos(sqrt(3) x)|  =  sqrt(3)
Let f(x) = 5 log(x):
Let v_1(x) = - integral (f(x) y_(b_2)(x))/((script capital w)(x)) dx and v_2(x) =  integral (f(x) y_(b_1)(x))/((script capital w)(x)) dx:
The particular solution will be given by:
y_p(x)  =  v_1(x) y_(b_1)(x)+v_2(x) y_(b_2)(x)
Compute v_1(x):
v_1(x)  =  - integral (5 log(x) sin(sqrt(3) x))/sqrt(3) dx  =  -5/3 (Ci(sqrt(3) x)-cos(sqrt(3) x) log(x))
Compute v_2(x):
v_2(x)  =   integral (5 cos(sqrt(3) x) log(x))/sqrt(3) dx  =  5/3 (log(x) sin(sqrt(3) x)-Si(sqrt(3) x))
The particular solution is thus:
y_p(x)  =  v_1(x) y_(b_1)(x)+v_2(x) y_(b_2)(x)  =  -5/3 cos(sqrt(3) x) (Ci(sqrt(3) x)-cos(sqrt(3) x) log(x))+5/3 (log(x) sin(sqrt(3) x)-Si(sqrt(3) x)) sin(sqrt(3) x)
Simplify:
y_p(x)  =  -5/3 (cos(sqrt(3) x) Ci(sqrt(3) x)-log(x)+sin(sqrt(3) x) Si(sqrt(3) x))
The general solution is given by:
Answer: | 
| y(x)  =  y_c(x) + y_p(x)  =  c_1 cos(sqrt(3) x)+c_2 sin(sqrt(3) x)-5/3 (cos(sqrt(3) x) Ci(sqrt(3) x)-log(x)+sin(sqrt(3) x) Si(sqrt(3) x))

Guest Nov 25, 2015
Sort: 

1+0 Answers

 #1
avatar
+15
Best Answer

y'' + 3y = 5lnx help?

This is a second-order linear ordinary differential equation. The solution to it is quite involved.

 

Solve ( d^2 y(x))/( dx^2)+3 y(x) = 5 log(x):
The general solution will be the sum of the complementary solution and particular solution.
Find the complementary solution by solving ( d^2 y(x))/( dx^2)+3 y(x)  =  0:
Assume a solution will be proportional to e^(lambda x) for some constant lambda.
Substitute y(x)  =  e^(lambda x) into the differential equation:
( d^2 )/( dx^2)(e^(lambda x))+3 e^(lambda x)  =  0
Substitute ( d^2 )/( dx^2)(e^(lambda x))  =  lambda^2 e^(lambda x):
lambda^2 e^(lambda x)+3 e^(lambda x)  =  0
Factor out e^(lambda x):
(lambda^2+3) e^(lambda x)  =  0
Since e^(lambda x) !=0 for any finite lambda, the zeros must come from the polynomial:
lambda^2+3  =  0
Solve for lambda:
lambda = i sqrt(3) or lambda = -i sqrt(3)
The roots lambda  =  ± i sqrt(3) give y_1(x) = c_1 e^(i sqrt(3) x), y_2(x) = c_2 e^(-i sqrt(3) x) as solutions, where c_1 and c_2 are arbitrary constants.
The general solution is the sum of the above solutions:
y(x)  =  y_1(x)+y_2(x)  =  c_1 e^(i sqrt(3) x)+c_2/e^(i sqrt(3) x)
Apply Euler's identity e^(alpha+i beta) = e^alpha cos(beta)+i e^alpha sin(beta):
y(x)  =  c_1 (cos(sqrt(3) x)+i sin(sqrt(3) x))+c_2 (cos(sqrt(3) x)-i sin(sqrt(3) x))
Regroup terms:
y(x)  =  (c_1+c_2) cos(sqrt(3) x)+i (c_1-c_2) sin(sqrt(3) x)
Redefine c_1+c_2 as c_1 and i (c_1-c_2) as c_2, since these are arbitrary constants:
y(x)  =  c_1 cos(sqrt(3) x)+c_2 sin(sqrt(3) x)
Determine the particular solution to ( d^2 y(x))/( dx^2)+3 y(x)  =  5 log(x) by variation of parameters:
List the basis solutions in y_c(x):
y_(b_1)(x) = cos(sqrt(3) x) and y_(b_2)(x) = sin(sqrt(3) x)
Compute the Wronskian of y_(b_1)(x) and y_(b_2)(x):
(script capital w)(x)  =  |cos(sqrt(3) x) | sin(sqrt(3) x)
( d)/( dx)(cos(sqrt(3) x)) | ( d)/( dx)(sin(sqrt(3) x))|  =  |cos(sqrt(3) x) | sin(sqrt(3) x)
-(sqrt(3) sin(sqrt(3) x)) | sqrt(3) cos(sqrt(3) x)|  =  sqrt(3)
Let f(x) = 5 log(x):
Let v_1(x) = - integral (f(x) y_(b_2)(x))/((script capital w)(x)) dx and v_2(x) =  integral (f(x) y_(b_1)(x))/((script capital w)(x)) dx:
The particular solution will be given by:
y_p(x)  =  v_1(x) y_(b_1)(x)+v_2(x) y_(b_2)(x)
Compute v_1(x):
v_1(x)  =  - integral (5 log(x) sin(sqrt(3) x))/sqrt(3) dx  =  -5/3 (Ci(sqrt(3) x)-cos(sqrt(3) x) log(x))
Compute v_2(x):
v_2(x)  =   integral (5 cos(sqrt(3) x) log(x))/sqrt(3) dx  =  5/3 (log(x) sin(sqrt(3) x)-Si(sqrt(3) x))
The particular solution is thus:
y_p(x)  =  v_1(x) y_(b_1)(x)+v_2(x) y_(b_2)(x)  =  -5/3 cos(sqrt(3) x) (Ci(sqrt(3) x)-cos(sqrt(3) x) log(x))+5/3 (log(x) sin(sqrt(3) x)-Si(sqrt(3) x)) sin(sqrt(3) x)
Simplify:
y_p(x)  =  -5/3 (cos(sqrt(3) x) Ci(sqrt(3) x)-log(x)+sin(sqrt(3) x) Si(sqrt(3) x))
The general solution is given by:
Answer: | 
| y(x)  =  y_c(x) + y_p(x)  =  c_1 cos(sqrt(3) x)+c_2 sin(sqrt(3) x)-5/3 (cos(sqrt(3) x) Ci(sqrt(3) x)-log(x)+sin(sqrt(3) x) Si(sqrt(3) x))

Guest Nov 25, 2015

7 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details