y" = sin(x)+e^(2x)
this should be no trouble. Just integrate twice and use the initial conditions to find the two constants.
f''(x)=sin(x)+e^(2x) f(0)=1/4, f'(0)=1/2
this should be no trouble. Just integrate twice and use the initial conditions to find the two constants.
Step 1: Solve the homogeneous equation.
The homogeneous equation is f''(x) = 0.
The solution to this equation is given by f(x) = Ax + B, where A and B are constants.
Step 2: Find the particular solution for the non-homogeneous equation.
The particular solution for the non-homogeneous equation f''(x) = sin(x) + e^(2x) can be found by using the method of undetermined coefficients.
Let's assume the particular solution has the form f_p(x) = C1*sin(x) + C2*cos(x) + C3*e^(2x), where C1, C2, and C3 are constants to be determined.
Taking the derivatives, we have:
f_p'(x) = C1*cos(x) - C2*sin(x) + 2*C3*e^(2x)
f_p''(x) = -C1*sin(x) - C2*cos(x) + 4*C3*e^(2x)
Plugging these derivatives back into the original differential equation, we get:
(-C1*sin(x) - C2*cos(x) + 4*C3*e^(2x)) = sin(x) + e^(2x)
Comparing the coefficients of each term on both sides, we can set up the following equations:
For sin(x): -C1 = 1
For cos(x): -C2 = 0
For e^(2x): 4*C3 = 1
Solving these equations, we find:
C1 = -1
C2 = 0
C3 = 1/4
Step 3: Combine the homogeneous and particular solutions.
The general solution to the non-homogeneous equation is given by:
f(x) = f_h(x) + f_p(x)
= Ax + B + (-1*sin(x) + 1/4*e^(2x))
Step 4: Apply the initial conditions to find the particular solution.
Using the initial conditions f(0) = 1/4 and f'(0) = 1/2, we can substitute these values into the general solution and solve for A and B.
f(0) = A*0 + B + (-1*sin(0) + 1/4*e^(2*0)) = B - sin(0) + 1/4*1 = B + 1/4 = 1/4
f'(0) = A + (-1*cos(0) + 1/4*2*e^(2*0)) = A - cos(0) + 1/2 = A - 1 + 1/2 = A + 1/2 = 1/2
Solving these two equations, we get:
B + 1/4 = 1/4 => B = 0
A + 1/2 = 1/2 => A = 0
Therefore, the particular solution of the differential equation that satisfies the given conditions is:
f(x) = 0*x + 0 + (-1*sin(x) + 1/4*e^(2x))
= -sin(x) + 1/4*e^(2x)
Step 1: Find the general solution of the homogeneous equation.
Since our equation is f''(x) = sin(x) + e^(2x), the homogeneous equation is f''(x) = 0. The general solution of the homogeneous equation is given by f(x) = Ax + B, where A and B are arbitrary constants.
Step 2: Find a particular solution.
To find a particular solution, we assume that the particular solution has the same form as the nonhomogeneous term, sin(x) + e^(2x). In this case, we assume that the particular solution has the form f_p(x) = C*sin(x) + D*e^(2x), where C and D are undetermined coefficients.
Step 3: Substitute the assumed particular solution into the original differential equation.
We differentiate the assumed particular solution twice (f_p''(x)) and substitute it into the original differential equation:
f_p''(x) = -C*sin(x) + 4D*e^(2x)
Substituting f_p''(x) into the original differential equation, we get:
-C*sin(x) + 4D*e^(2x) = sin(x) + e^(2x)
Step 4: Equate similar terms and solve for the undetermined coefficients.
Equate the coefficients of sin(x) and e^(2x) on both sides of the equation. This gives us two equations:
-C = 1 (coefficient of sin(x))
4D = 1 (coefficient of e^(2x))
Solving these equations, we find:
C = -1 and D = 1/4
Step 5: Use the particular solution and the general solution of the homogeneous equation to find the complete solution.
The particular solution is f_p(x) = -sin(x) + (1/4)*e^(2x)
The general solution of the homogeneous equation is f(x) = Ax + B
The complete solution is given by the sum of the particular solution and the general solution of the homogeneous equation:
f(x) = Ax + B - sin(x) + (1/4)*e^(2x)
Step 6: Apply the initial conditions to find the values of A and B.
Given f(0) = 1/4 and f'(0) = 1/2, we can substitute these values into the complete solution and solve for A and B.
f(0) = A(0) + B - sin(0) + (1/4)*e^(2(0)) = B + (1/4) = 1/4
Solving this equation, we find B = 0.
Differentiating the complete solution, we get:
f'(x) = A - cos(x) + (1/2)*e^(2x)
f'(0) = A - cos(0) + (1/2)*e^(2(0)) = A - 1 + (1/2) = 1/2
Solving this equation, we find A = 1.
Step 7: Write the particular solution that satisfies the given conditions.
Substituting the values of A and B into the complete solution, we get:
f(x) = x - sin(x) + (1/4)*e^(2x)
Therefore, the particular solution of the given differential equation that satisfies the conditions f(0) = 1/4 and f'(0) = 1/2 is f(x) = x - sin(x) + (1/4)*e^(2x).