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can you explain each step thanks

(a) Find the general solution of the the second-order linear ODE \[ 2 y^{\prime \prime}-5 y^{\prime}-3 y=0 . \] (b) Find the general solution of the inhomogeneous second-order linear ODE \[ 2 y^{\prime \prime}-5 y^{\prime}-3 y=10 \sin x . \] (c) Find the solution to the Initial Value Problem \[ 2 y^{\prime \prime}-5 y^{\prime}-3 y=10 \sin x, \quad y(0)=4, \quad y^{\prime}(0)=1 . \] Question 2 [26 marks]. (a) Check whether the IVP \[ y^{\prime}=\frac{x}{y-4}, \quad y(0)=4 . \] satisfies the hypotheses of the Picard-Lindelöf theorem. (b) Find all the solutions of the IVP defined in (a). Is this result in contradiction with the result obtained in (a)? Explain your answer. (c) Determine the smallest \( b>0 \) such that the BVP \[ 2 y^{\prime \prime}-18 y=\tanh (x), \quad y(0)=0, y^{\prime}(b)=3, \] does not have a unique solution.

2y??5y??3y=0 Rewrite the differential equation. 2d2ydx2?5dydx?3y=0 Assume all solutions are of the form y=erx. Find the characteristic equation for 2d