solve ivp y''+4y=2x ,y(0)=1,y'(0)=2

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solve ivp y''+4y=2x ,y(0)=1,y'(0)=2

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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For the complimentary solution (i.e. solving y''+4y = 0) which you'll need you can follow the basic process outlined at http://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx Then for the particular solution you can use either Undetermined Coefficients (probably easiest for this problem) at, http://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx Or variation of parameters at http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx. The first and second example on the Undetermined Coefficients page shows how to solve a full IVP using this process.

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