Differential Equations

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Question

Solve the differential equation:
$dy over dx minus 2 straight y space equals space 3 straight x.$

Solution

The given differential equation is

dy over dx minus 2 straight y space equals space 3 straight x

Comparing it with

dy over dx plus space straight P space straight y space equals space straight Q comma space space we space get comma

P = -2, Q = 3x

integral straight P space dx space equals space minus 2 space integral space dx space equals space minus 2 straight x space space space straight I. straight F. space equals space straight e to the power of integral Pdx end exponent space equals space straight e to the power of negative 2 straight x end exponent

Solution of given differential equation is

straight y space straight e to the power of integral straight P space dx end exponent space equals space integral straight Q space straight e to the power of integral straight P space dx end exponent dx space plus space straight c

straight y space straight e to the power of negative 2 straight x end exponent space equals space integral space 3 space straight x space straight e to the power of negative 2 straight x end exponent space dx plus straight c

ye to the power of negative 2 straight x end exponent space equals space 3 space integral space xe to the power of negative 2 straight x end exponent dx plus straight c

ye to the power of negative 2 straight x end exponent space equals space 3 space open square brackets straight x fraction numerator straight e to the power of negative 2 straight x end exponent over denominator negative 2 end fraction minus integral 1. space fraction numerator straight e to the power of negative 2 straight x end exponent over denominator negative 2 end fraction dx close square brackets plus straight c

ye to the power of negative 2 straight x end exponent space equals negative 3 over 2 xe to the power of negative 2 straight x end exponent plus 3 over 2 integral straight e to the power of negative 2 end exponent dx plus straight c

ye to the power of negative 2 straight x end exponent space equals space minus 3 over 2 xe to the power of negative 2 straight x end exponent plus 3 over 2 fraction numerator straight e to the power of negative 2 straight x end exponent over denominator negative 2 end fraction plus straight c

straight y equals negative 3 over 2 straight x minus 3 over 4 plus ce to the power of 2 straight x end exponent

which is required solution.