Final answer:
To find the general solution to the differential equation yʹ'(θ)+25 y(θ)=2 ³ 5 θ, we can use the method of integrating factors. Multiply both sides of the equation by the integrating factor e^(25θ), and then integrate both sides with respect to θ to find the general solution.
Step-by-step explanation:
To find the general solution to the differential equation yʹ'(θ)+25 y(θ)=2 ³ 5 θ, we can first rewrite the equation as yʹ'(θ) = 2 ³ 5 θ - 25 y(θ). This is a linear first-order ordinary differential equation. To solve it, we can use the method of integrating factors. Multiply both sides of the equation by the integrating factor e^(25θ): e^(25θ)yʹ' + 25e^(25θ)y = 2 ³ 5 θe^(25θ). Notice that the left side of the equation is the derivative of (e^(25θ)y) with respect to θ. So we can rewrite the equation as (e^(25θ)y)' = 2 ³ 5 θe^(25θ). Integrate both sides with respect to θ to get (e^(25θ)y) = ∫(2 ³ 5 θe^(25θ)) dθ. Solve the integral and divide by e^(25θ) to find the general solution: y(θ) = (∫(2 ³ 5 θe^(25θ)) dθ) / e^(25θ).