Final answer:
To find the general solution of the homogeneous differential equation x''+7x'+3x=0 with initial conditions x(2)=1 and x'(2)=2, solve the characteristic quadratic equation for r, write the general solution using the roots, and then apply initial conditions to determine the constants.
Step-by-step explanation:
The student is asking for the general solution of a second-order homogeneous linear differential equation with constant coefficients, given by x''+7x'+3x=0.
To solve this, one must first find the characteristic equation, which is obtained by replacing x with ert, leading to a quadratic, characteristic equation r2+7r+3=0. Solving this using the quadratic formula will give the roots (eigenvalues). From these roots, we can set up the general solution of the form C1er1t+C2er2t. By applying the initial conditions x(2)=1 and x'(2)=2, we can solve for the constants C1 and C2.
Using the quadratic formula, we solve the characteristic equation to find the values for r. Once the roots are found, the general solution can be written down and then we use the initial conditions to find the specific values for C1 and C2 that satisfy the initial conditions. This will give us the particular solution of the differential equation for the given initial conditions.