Final answer:
To solve the initial value system {5x+(D+3)y=-1, (D+2)x+(D+1)y=0} with initial values x(0)=1 and y(0)=-2, we can use the method of substitution. By substituting the given initial values and using algebraic manipulation, we can find the values of x and y that satisfy the system of equations. The solution to the system of equations is x=1/2 and y=2.
Step-by-step explanation:
To solve the initial value system {5x+(D+3)y=-1, (D+2)x+(D+1)y=0} with initial values x(0)=1 and y(0)=-2, we can use the method of substitution.
- 1. Solve the second equation for x using the initial values y(0)=-2 and x(0)=1. Substitute y=-2 and x=1 into the second equation and solve for x: (D+2)(1)+(D+1)(-2)=0. Simplify the equation to get: D+2-2D-2=0
- Substitute the values of x and y from step 1 into the first equation: 5(1)+(D+3)(-2)=-1. Simplify the equation to get: 5-2D-6=-1
- Solve the resulting equation for D to find its value. Simplify the equation to get: -2D-1=0. Solve for D to get D=-1/2
- Substitute the value of D into the second equation to find y. Simplify the equation to get: (-1/2+2)(1)+(D+1)(y)=0. Solve for y to get y=2
- Finally, substitute the values of D, x, and y into the first equation to find x. Simplify the equation to get: 5x+(-1/2+3)(2)=-1. Solve for x to get x=1/2