solve (1+x)^2d^y/dx^2+(1+x)dy/dx+y=4cosh(x+1)
To solve the given differential equation, we can use the method of undetermined coefficients. First, let's simplify the equation:
(1 + x)^2(d^2y/dx^2) + (1 + x)(dy/dx) + y = 4cosh(x + 1)
Let's assume a particular solution in the form of y_particular = A*cosh(x + 1), where A is a constant to be determined.
Now, we will find the first and second derivatives of y_particular:
dy_particular/dx = A*sinh(x + 1)
d^2y_particular/dx^2 = A*cosh(x + 1)
Substituting these values back into the differential equation:
(1 + x)^2(A*cosh(x + 1)) + (1 + x)(A*sinh(x + 1)) + A*cosh(x + 1) = 4cosh(x + 1)
Simplifying this equation, we have:
(A*(1 + x)^2 + A*(1 + x) + A)*cosh(x + 1) = 4cosh(x + 1)
Now, equating the coefficients of cosh(x + 1) on both sides of the equation:
A*(1 + x)^2 + A*(1 + x) + A = 4
Expanding (1 + x)^2 and simplifying further:
A*(1 + 2x + x^2) + A*(1 + x) + A = 4
Simplifying and collecting like terms:
A*x^2 + 3A*x + 3A = 4
Now, we can equate the coefficients on both sides of the equation:
A = 0 (coefficient of x^2 term)
3A = 0 (coefficient of x term)
3A = 4 (constant term)
From these equations, we can see that A = 0, which means that there is no particular solution for y.
Therefore, the general solution for the given differential equation is y_general = y_homogeneous + y_particular.
To find the homogeneous solution, let's assume y_homogeneous = e^(mx). Differentiating twice with respect to x, we have:
dy_homogeneous/dx = me^(mx)
d^2y_homogeneous/dx^2 = m^2*e^(mx)
Substituting these values back into the differential equation:
(1 + x)^2(m^2*e^(mx)) + (1 + x)*(me^(mx)) + e^(mx) = 0
Expanding and simplifying, we have:
m^2*(1 + x)^2*e^(mx) + m*(1 + x)*e^(mx) + e^(mx) = 0
Factoring out e^(mx), we get:
e^(mx)*(m^2*(1 + x)^2 + m*(1 + x) + 1) = 0
Since e^(mx) is never equal to zero, the expression inside the parentheses must be equal to zero:
m^2*(1 + x)^2 + m*(1 + x) + 1 = 0
To solve this quadratic equation for m, you can use the quadratic formula:
m = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = 1, b = 1 + x, and c = 1. Substitute these values into the quadratic formula and solve for m.
Once you have the values of m, the homogeneous solution will be y_homogeneous = C1*e^(m1*x) + C2*e^(m2*x), where C1 and C2 are constants determined by any initial conditions given in the problem.