Find the general solution of this de (1+cosx)y'=sinx(sinx+sinxcosx-y)
I need full step have tried but nothing good is coming out
Rearrange things into the usual form for a 1st-order linear DE:
(1+cosx)y' = sin^2x(1+cosx) - sinx y
y' - sinx/(1+cosx) y = sin^2x
y' - tan(x/2) y = sin^2x
Now you have the form y' + p(x) y = q(x)
The integrating factor is now e^∫p(x) dx = 1/2 sec^2(x/2)
1/2 sec^2(x/2) y' - tan(x/2)(1/2 sec^2(x/2))y = sin^2(x) * 1/2 sec^2(x/2)
After some trig manipulation, this comes out to be
(tan(x/2) y)' = 4(1+cos^2(x))/(1+cos(x))
Some more study should provide the way to integrate the right side.
Check my math, and don't be afraid to use tools like wolframalpha to provide some help.
To find the general solution of the given differential equation:
(1 + cos(x))y' = sin(x)(sin(x) + sin(x)cos(x) - y)
Let's go through the steps:
Step 1: Rearrange the equation
Multiply both sides by (1 + cos(x)):
y' = sin(x)(sin(x) + sin(x)cos(x) - y)/(1 + cos(x))
Step 2: Simplify the expression
Distribute sin(x) into the brackets:
y' = sin^2(x) + sin^2(x)cos(x) - ysin(x)/(1 + cos(x))
Step 3: Rewrite sin^2(x) using trigonometric identities
Recall that sin^2(x) = 1 - cos^2(x):
y' = (1 - cos^2(x)) + sin^2(x)cos(x) - ysin(x)/(1 + cos(x))
Step 4: Combine like terms
y' = 1 - cos^2(x) + sin^2(x)cos(x) - ysin(x)/(1 + cos(x))
Step 5: Simplify the equation further
Using the identity sin^2(x) = 1 - cos^2(x) and simplifying the expression:
y' = 1 + (1 - cos^2(x))cos(x) - ysin(x)/(1 + cos(x))
y' = 1 + cos(x) - cos^3(x) - ysin(x)/(1 + cos(x))
Step 6: Separate the variables
Rearrange the equation to have all terms involving 'y' on one side and all terms involving 'x' on the other side:
y' + ysin(x)/(1 + cos(x)) = 1 + cos(x) - cos^3(x)
Step 7: Solve the homogeneous part
The homogeneous part is y' + ysin(x)/(1 + cos(x)) = 0
This can be solved by separation of variables and integrating:
∫(dy/y) = -∫sin(x)/(1 + cos(x)) dx
Integrating the left side gives:
ln|y| = -ln|1 + cos(x)| + C1 (where C1 is the constant of integration)
Step 8: Solve the particular part
To find a particular solution, we can use the method of undetermined coefficients.
Assuming y = A + Bcos(x) + Csin(x) will satisfy the particular solution, where A, B, and C are constants:
Taking the derivatives, we have:
y' = -Bsin(x) + Ccos(x)
Substituting y and y' back into the original differential equation, we get:
(-Bsin(x) + Ccos(x)) + (A + Bcos(x) + Csin(x))(sin(x)/(1 + cos(x))) = 1 + cos(x) - cos^3(x)
Simplifying the equation and comparing coefficients, we find:
-C = 1,
-Asin(x) - Bcos(x) + Csin(x)/(1 + cos(x)) = cos(x) - cos^3(x),
B = 0.
From the first equation, we have C = -1.
Substituting this value of C into the second equation, we have:
-Asin(x) + sin(x)/(1 + cos(x)) = cos(x) - cos^3(x)
Simplifying further and solving for A, we get:
A = -cos(x) + cos^3(x) - sin(x)/(1 + cos(x))
Therefore, the particular solution is:
y_particular = -cos(x) + cos^3(x) - sin(x)/(1 + cos(x))
Step 9: General solution
Now that we have the homogeneous solution and the particular solution, the general solution of the given differential equation is given by:
y = e^(ln|1 + cos(x)| - C1) * (A + Bcos(x) + Csin(x)) + y_particular
Simplifying further:
y = (1 + cos(x)) * (A + Bcos(x) + Csin(x)) + y_particular
where A = -cos(x) + cos^3(x) - sin(x)/(1 + cos(x)), B = 0, C = -1, and C1 is the constant of integration.
To solve the given differential equation, we can follow these steps:
Step 1: Rewrite the equation in a more standard form.
Step 2: Find the integrating factor.
Step 3: Integrate both sides of the equation.
Step 4: Solve for the constant of integration.
Step 5: Write down the general solution.
Let's go through each step in detail:
Step 1: Rewrite the equation in a more standard form.
The given differential equation is: (1 + cos(x))y' = sin(x)(sin(x) + sin(x)cos(x) - y)
Step 2: Find the integrating factor.
The integrating factor, denoted by μ(x), is obtained by multiplying the coefficient of y' by an integrating factor function. In this case, the coefficient of y' is (1 + cos(x)).
μ(x) = e^(∫ (1 + cos(x)) dx) (where ∫ denotes integration)
Simplifying, we have:
μ(x) = e^x * e^(∫ cos(x) dx)
Integrating cos(x), we get:
μ(x) = e^x * (sin(x) + C1), where C1 is the constant of integration
Step 3: Integrate both sides of the equation.
Multiply both sides of the given differential equation by the integrating factor:
e^x * (sin(x) + C1) * (1 + cos(x))y' = e^x * (sin(x) + C1) * sin(x)(sin(x) + sin(x)cos(x) - y)
Integrating both sides separately yields:
∫ [e^x * (sin(x) + C1) * (1 + cos(x))y'] dx = ∫ [e^x * (sin(x) + C1) * sin(x)(sin(x) + sin(x)cos(x) - y) dx]
On the left side, use the substitution u = (sin(x) + C1) * (1 + cos(x)), and then du = [sin(x) + C1 + cos(x)sin(x) + C1cos(x)] dx.
The integral then becomes:
∫ e^x * y' du
On the right side, simplify the expression and perform the appropriate integrations.
Step 4: Solve for the constant of integration.
Integrate the expression on the left side to obtain:
y = ∫ e^x * y' du + C2, where C2 is the constant of integration
Step 5: Write down the general solution.
Finally, the general solution of the given differential equation is:
y = e^x * ∫ e^(-x) * [sin(x) + C1 + sin(x)cos(x) + C1cos(x)] du + C2
Now, you can simplify the integral and constants to obtain a more concise form for the general solution.