Asked by xX_Supaman_Xx
Solve the following differential equation, i.e. solve for y:
dy/dx - y/x = 1/x + y + 1
I have this so far,
1. dy/dx= (y + 1 + xy + x)/x
2. x dy = (y + 1 + xy + x)dx
3. Integrate both sides...
xy + c = xy + x + .5x^2y + .5x^2 + k
where c and k are constants
4. c = x + .5x^2 (y + 1) + k
5. .5x^2 (y + 1) = c - k -x
6. y = 2(C- x)x^-2 - 1
where C = c - k
but I have been told this is wrong. so can soneone please direct me in the error of my method and show me the correct working out.
dy/dx - y/x = 1/x + y + 1
I have this so far,
1. dy/dx= (y + 1 + xy + x)/x
2. x dy = (y + 1 + xy + x)dx
3. Integrate both sides...
xy + c = xy + x + .5x^2y + .5x^2 + k
where c and k are constants
4. c = x + .5x^2 (y + 1) + k
5. .5x^2 (y + 1) = c - k -x
6. y = 2(C- x)x^-2 - 1
where C = c - k
but I have been told this is wrong. so can soneone please direct me in the error of my method and show me the correct working out.
Answers
Answered by
drwls
What is wrong is this: In step 3, you cannot treat x as a constant in the dy integration (on the left), nor can your treat y as a constant in the dy integration (on the right). You need to totally separate y and x variables on opposite sides somehow, or find some other trick.
This is a first order linear differential equation that can be written as
dy/dx - Py = Q
where P and Q are functions of x.
Equations of this type can be solved by the "integrating factor" method. I suggest you familiarize yourself with the method. It goes like this:
Compute the function
rho(x) = exp(integral of P dx)
The solution will be
rho*y = [Integral of rho*Q(x)dx] + C
This is a first order linear differential equation that can be written as
dy/dx - Py = Q
where P and Q are functions of x.
Equations of this type can be solved by the "integrating factor" method. I suggest you familiarize yourself with the method. It goes like this:
Compute the function
rho(x) = exp(integral of P dx)
The solution will be
rho*y = [Integral of rho*Q(x)dx] + C
Answered by
drwls (correction)
I should have written the standard form as dy/dx + P(x)y = Q(x)
I got the sign wrong on the P.
In your case, Q = -P = (1/x) + 1
The form of the solution which I wrote should be correct.
I got the sign wrong on the P.
In your case, Q = -P = (1/x) + 1
The form of the solution which I wrote should be correct.
Answered by
drwls
For the first step, I get
rho(x) = 1/(x*e^x)
The final step would be to solve
y(x) = (x*e^x)*(Integral)[(1/x^2)*e^-x + (1/x)*e^-x] dx + C
rho(x) = 1/(x*e^x)
The final step would be to solve
y(x) = (x*e^x)*(Integral)[(1/x^2)*e^-x + (1/x)*e^-x] dx + C
Answered by
Damon
Yes, I worked it through mostly the same way and got
y = c x e^x - 1
y = c x e^x - 1
Answered by
Damon
By the way, having gotten interested, I also solved this by brute force trial and error.
I assumed an exponential type solution:
y = Ae^x + Bxe^x+ C
then
dy/dx = Ae^x + Bxe^x + Be^x
plug that in
Ae^x + Bxe^x + Be^x - Ae^x/x - Be^x - C/x = 1/x + 1 + Ae^x + Bxe^x+ C
combine like terms
-Ae^x/x - C/x = 1/x + 1+ C
then
-Ae^x -C = 1 + x + Cx
well now, A = 0 because we have no other term in e^x
C(x+1)= -(x+1)
so
C = -1
B can be any constant
so
y = B e^x - 1
same answer
I assumed an exponential type solution:
y = Ae^x + Bxe^x+ C
then
dy/dx = Ae^x + Bxe^x + Be^x
plug that in
Ae^x + Bxe^x + Be^x - Ae^x/x - Be^x - C/x = 1/x + 1 + Ae^x + Bxe^x+ C
combine like terms
-Ae^x/x - C/x = 1/x + 1+ C
then
-Ae^x -C = 1 + x + Cx
well now, A = 0 because we have no other term in e^x
C(x+1)= -(x+1)
so
C = -1
B can be any constant
so
y = B e^x - 1
same answer
Answered by
Damon
Whoops, correct last line
y = B xe^x - 1
y = B xe^x - 1