Question

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

Answer 1

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.