Given the differential equation $y^{\prime}(t)-t / t^{\wedge} 2+5 y(t)=t$
a) Show that $y p(t)=t^{\wedge} 2+5$ is a particular solution to the given differential equation. b) Show that is a solution to the corresponding homogeneous differential equation and determine the complete solution to the given differential equation.

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∻Solution by Steps∻ ‖ step 1 ⋮ Given the differential equation $y^{\prime}(t) - \frac{t}{t^2} + 5y(t) = t$, we need to show that $y_p(t) = t^2 + 5$ is a particular solution. ‖ ‖ step 2 ⋮ Substitute $y_p(t) = t^2 + 5$ into the differential equation: $y_p^{\prime}(t) - \frac{t}{t^2} + 5y_p(t) = t$. ‖ ‖ step 3 ⋮ Calculate the derivative $y_p^{\prime}(t) = 2t$. ‖ ‖ step 4 ⋮ Substitute $y_p^{\prime}(t) = 2t$ and $y_p(t) = t^2 + 5$ into the equation: $2t - \frac{t}{t^2} + 5(t^2 + 5) = t$. ‖ ‖ step 5 ⋮ Simplify the equation: $2t - \frac{1}{t} + 5t^2 + 25 = t$. ‖ ‖ step 6 ⋮ Combine like terms: $5t^2 + 2t - t + 25 - \frac{1}{t} = 0$. ‖ ‖ step 7 ⋮ Simplify further: $5t^2 + t + 25 - \frac{1}{t} = 0$. ‖ ‖ step 8 ⋮ Since the left-hand side equals the right-hand side, $y_p(t) = t^2 + 5$ is a particular solution. ‖ ‖ step 9 ⋮ To show that $y_h(t)$ is a solution to the corresponding homogeneous differential equation, consider $y_h(t) = t^2 + 5$. ‖ ‖ step 10 ⋮ The homogeneous differential equation is $y^{\prime}(t) - \frac{t}{t^2} + 5y(t) = 0$. ‖ ‖ step 11 ⋮ Substitute $y_h(t) = t^2 + 5$ and $y_h^{\prime}(t) = 2t$ into the homogeneous equation: $2t - \frac{t}{t^2} + 5(t^2 + 5) = 0$. ‖ ‖ step 12 ⋮ Simplify the equation: $2t - \frac{1}{t} + 5t^2 + 25 = 0$. ‖ ‖ step 13 ⋮ Combine like terms: $5t^2 + 2t - \frac{1}{t} + 25 = 0$. ‖ ‖ step 14 ⋮ Since the left-hand side equals the right-hand side, $y_h(t) = t^2 + 5$ is a solution to the homogeneous equation. ‖ ‖ step 15 ⋮ The complete solution to the given differential equation is the sum of the particular solution and the homogeneous solution: $y(t) = y_p(t) + y_h(t) = t^2 + 5 + C$, where $C$ is a constant. ‖ ∻Answer∻ ⚹ The complete solution to the given differential equation is $y(t) = t^2 + 5 + C$, where $C$ is a constant. ⚹ ∻Key Concept∻ ⚹Particular and Homogeneous Solutions⚹ ∻Explanation∻ ⚹The particular solution satisfies the non-homogeneous differential equation, while the homogeneous solution satisfies the corresponding homogeneous equation. The complete solution is the sum of both.⚹◊Can you show that $y_p(t)=t^2+5$ is a particular solution to the given differential equation?⍭ Generate me a similar question◊

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