Solve the differential equation y\" + 6 y\' + 9 y = 0 . Give the most general so

Question

Solve the differential equation y" + 6 y' + 9 y = 0 . Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. y(x) = Solve the differential equation /" + 3 + 2 / = 0 with the initial conditions y(0) = 1 , y'(0)=l . Write y as a real-valued function of x. y(x) = Solve the differential equation y" + 6 y' + 25 y = 0 with the initial conditions y(0) = 3 , y'(0)=3 . Write y as a real-valued function of x. Submit Answer Save Progress Solve the differential equation y" - y' -56 y = 0 with the initial conditions y(0) = 4 , y'(0)=-43 Write y as a real-valued function of x. Solve the differential equation y" -10 y' + 16 y = 0 . Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. Solve the differential equation /" + 29 / = 0 . Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. y(x) = Solve the differential equation y" -5 = 0 . Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. y(x) = Solve the differential equation y" + 5 y' + 4 y = 0 . Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. y(x) = Solve the differential equation y" - 2 y = 0 . Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. y(x) = Submit Answer Save Progress Write the number in polar form (that is, in the form r(cos0 + i sin0)) with argument between 0 and 2n. Solve the differential equation y" + 6 y' + 9 y = 0 . Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. y(x) = Solve the differential equation /" + 3 + 2 / = 0 with the initial conditions y(0) = 1 , y'(0)=l . Write y as a real-valued function of x. Solve the differential equation y" + 6 y' + 25 y = 0 with the initial conditions y(0) = 3 , y'(0)=3 . Write y as a real-valued function of x. Submit Answer Save Progress Solve the differential equation y" - y' -56 y = 0 with the initial conditions y(0) = 4 , y'(0)=-43 Write y as a real-valued function of x. 5. H+ -/1 points Solve the differential equation y" -10 y' + 16 y = 0 . Give the most general solution, using A and B for any unknown constants, and write y as a real-valued function of x. Write the number in polar form (that is, in the form r(cos0 + i sin0)) with argument between 0 and pi. Write the number in polar form (that is, in the form r(cos0 + i sin0 ) with argument between 0 and 2pi. Simplify the following power of a complex number. (Hint: use Euler's formula) Simplify the following power of a complex number. (Hint: use Euler's formula) Solve the differential equation xy' = y + xe2yfx by making the change of variable v = y/x. Use a capital C for any needed unknown constant in your answer.

Explanation / Answer

Solve 3 ( dy(x))/( dx)+( d^2 y(x))/( dx^2)+2 y(x) = 0, such that y(0) = 1 and y'(0) = 1: Assume a solution will be proportional to e^(lambda x) for some constant lambda. Substitute y(x) = e^(lambda x) into the differential equation: ( d^2 )/( dx^2)(e^(lambda x))+3 ( d)/( dx)(e^(lambda x))+2 e^(lambda x) = 0 Substitute ( d^2 )/( dx^2)(e^(lambda x)) = lambda^2 e^(lambda x) and ( d)/( dx)(e^(lambda x)) = lambda e^(lambda x): lambda^2 e^(lambda x)+3 lambda e^(lambda x)+2 e^(lambda x) = 0 Factor out e^(lambda x): (lambda^2+3 lambda+2) e^(lambda x) = 0 Since e^(lambda x) !=0 for any finite lambda, the zeros must come from the polynomial: lambda^2+3 lambda+2 = 0 Factor: (lambda+1) (lambda+2) = 0 Solve for lambda: lambda = -2 or lambda = -1 The root lambda = -2 gives y_1(x) = c_1 e^(-2 x) as a solution, where c_1 is an arbitrary constant. The root lambda = -1 gives y_2(x) = c_2 e^(-x) as a solution, where c_2 is an arbitrary constant. The general solution is the sum of the above solutions: y(x) = y_1(x)+y_2(x) = c_1/e^(2 x)+c_2/e^x Solve for the unknown constants using the initial conditions: Compute ( dy(x))/( dx): ( dy(x))/( dx) = ( d)/( dx)(c_1/e^(2 x)+c_2/e^x) = -(2 c_1)/e^(2 x)-c_2/e^x Substitute y(0) = 1 into y(x) = c_1 e^(-2 x)+c_2 e^(-x): c_1+c_2 = 1 Substitute y'(0) = 1 into ( dy(x))/( dx) = -2 c_1 e^(-2 x)-c_2 e^(-x): -2 c_1-c_2 = 1 Solve the system: c_1 = -2 c_2 = 3 Substitute c_1 = -2 and c_2 = 3 into y(x) = c_1 e^(-2 x)+c_2 e^(-x): Answer: | | y(x) = (3 e^x-2)/e^(2 x)

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.