What is a solution to the differential equation xy"+2y'?

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Answer 1 · 2018-02-25T09:03:30

#y=C_2/x+C_3#

Assuming we want to solve the differential equation

#xy''+2y'=0#

Let #z=y'# then #z'=y''#

#xz'+2z=0#

Now we have a first order differential equation,
which can be solved using seperation of variables

#x(dz)/dx+2z=0=>1/(2z)dz=-1/xdx#

Integrate both sides

#1/2int1/zdz=-int1/xdx#

#=>1/2ln(z)=-ln(x)+C#

#=>z=C_1/x^2#

Substitute back #z=y'#

#y'=C_1/x^2=>y=intC_1/x^2dx=>y=C_2/x+C_3#