Maths Assignment help
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The document provides solutions to various differential equations, including first-order and second-order linear homogeneous and non-homogeneous cases. It details methods such as the characteristic equation and undetermined coefficients to derive general and particular solutions. Key results include solutions for equations like (dy/dx) + 2y = 0 and (d^2y/dx^2) - 4y = 0, among others.
Maths Assignment help
- 1. Differential Equation Reach Out To Us - Address: Kolb Rd, Tucson, Arizona, USA Call Us: +1 (315) 557-6473 Email: info@mathsassignmenthelp.com
- 2. www.mathsassignmenthelp.com The differential equation (dy/dx) + 2y = 0 has the general solution: a) y = C1 + 2x b) y = C1 - 2x c) y = C1e^(-2x) d) y = C1e^(2x) Solution: The differential equation (dy/dx) + 2y = 0 is a first-order linear homogeneous differential equation. Its general solution is given by y = C1e^(-2x), so the answer is c. The differential equation (d^2y/dx^2) - 4y = 0 has the general solution: a) y = C1e^(-2x) + C2e^(2x) b) y = C1sin(2x) + C2cos(2x) c) y = C1e^(2x) + C2xe^(2x) d) y = C1sinh(2x) + C2cosh(2x) Solution: The differential equation (d^2y/dx^2) - 4y = 0 is a second-order linear homogeneous differential equation with constant coefficients. Its characteristic equation is r^2 - 4 = 0, which has roots r = ±2. Therefore, the general solution is y = C1e^(2x) + C2e^(-2x), so the answer is a.
- 3. www.mathsassignmenthelp.com The differential equation (dy/dx) + y = 3 has the particular solution: a) y = 2e^x + 1 b) y = 3e^x + 2 c) y = 4e^x + 3 d) y = 5e^x + 4 Solution: The differential equation (dy/dx) + y = 3 is a first-order linear non-homogeneous differential equation. Its homogeneous part is (dy/dx) + y = 0, which has the general solution y = C1e^(-x). To find the particular solution, we can use the method of undetermined coefficients. Assuming that the particular solution is of the form y = Ae^x + B, we get (d/dx)(Ae^x + B) + Ae^x + B = 3. Simplifying this equation, we get A = 2 and B = 1, so the particular solution is y = 2e^x + 1, and the answer is a. The differential equation (d^2y/dx^2) - 5(dy/dx) + 6y = 0 has the general solution: a) y = (C1 + C2x)e^2x b) y = C1e^2x + C2e^3x c) y = C1e^2x + C2e^x d) y = (C1 + C2x)e^x Solution: The differential equation (d^2y/dx^2) - 5(dy/dx) + 6y = 0 is a second-order linear homogeneous differential equation with constant coefficients. Its characteristic equation is r^2 - 5r + 6 = 0, which factors as (r - 2)(r - 3) = 0. Therefore, the general solution is y = C1e^(2x) + C2e^(3x), and the answer is b.
- 4. www.mathsassignmenthelp.com The differential equation (dy/dx) = 2 has the general solution: a) y = 2x + C1 b) y = 2x + C2 c) y = 2 d) y = 2x Solution: The differential equation (dy/dx) = 2 is a first-order separable differential equation. Integrating both sides with respect to x, we get y = 2x + C, where C is the constant of integration. Therefore, the general solution is y = 2x + C1, and the answer is a. The differential equation (d^2y/dx^2) + 4(dy/dx) + 4y = 0 has the general solution: a) y = C1e^(-2x) + C2xe^(-2x) b) y = C1sin(2x) + C2cos(2x) c) y = C1e^(-2x) + C2e^(2x) d) y = C1sinh(2x) + C2cosh(2x) Solution: The differential equation (d^2y/dx^2) + 4(dy/dx) + 4y = 0 is a second-order linear homogeneous differential equation with constant coefficients. Its characteristic equation is r^2 + 4r + 4 = 0, which has a double root at r = -2. Therefore, the general solution is y = (C1 + C2x)e^(-2x), and the answer is a.
- 5. www.mathsassignmenthelp.com The differential equation (d^2y/dx^2) + y = 0 has the general solution: a) y = C1sin(x) + C2cos(x) b) y = C1e^x + C2e^(-x) c) y = C1sinh(x) + C2cosh(x) d) y = C1 + C2x Solution: The differential equation (d^2y/dx^2) + y = 0 is a second-order linear homogeneous differential equation with constant coefficients. Its characteristic equation is r^2 + 1 = 0, which has roots r = ±i. Therefore, the general solution is y = C1sin(x) + C2cos(x), and the answer is a. The differential equation (d^2y/dx^2) - 2(dy/dx) + y = 0 has the general solution: a) y = (C1 + C2x)e^x b) y = C1e^x + C2e^(2x) c) y = C1e^(-x) + C2xe^(-x) d) y = (C1 + C2x)e^(-x) Solution: The differential equation (d^2y/dx^2) - 2(dy/dx) + y = 0 is a second-order linear homogeneous differential equation with constant coefficients. Its characteristic equation is r^2 - 2r + 1 = 0, which has a double root at r = 1. Therefore, the general solution is y = (C1 + C2x)e^x, and the answer is a.