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Differential Equations

Question

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Solve:
$dy over dx plus 4 straight y space equals space 5 straight x$

Solution

The given differential equation is

dy over dx plus 4 straight y space equals space 5 straight x

Comparing it with

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

we get,
P = 4, Q = 5x

integral straight P space dx space equals space integral 4 space dx space equals space 4 straight x

straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of 4 straight x end exponent

Solution of given differential equation is

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

or

ye to the power of 4 straight x end exponent space equals space integral 5 space straight x space straight e to the power of 4 straight x end exponent dx space plus straight c

or space space space space space straight y space straight e to the power of 4 straight x end exponent space equals space 5 space integral straight x space straight e to the power of 4 straight x end exponent dx plus straight c or space space space space straight y space straight e to the power of 4 straight x end exponent space equals space 5 space open square brackets straight x space straight e to the power of 4 straight x end exponent over 4 minus space integral 1. space straight e to the power of 4 straight x end exponent over 4 dx close square brackets plus straight c or space space space space straight y space straight e to the power of 4 straight x end exponent space equals space 5 over 4 straight x space straight e to the power of 4 straight x end exponent minus 5 over 4 integral straight e to the power of 4 straight x end exponent dx space plus space straight c or space space space straight y space straight e to the power of 4 straight x end exponent space equals space 5 over 4 straight x space straight e to the power of 4 straight x end exponent space minus space 5 over 4. space straight e to the power of 4 straight x end exponent over 4 plus straight c therefore space space space space space space space space straight y space equals space 5 over 4 straight x minus 5 over 16 plus ce to the power of negative 4 straight x end exponent

is required solution.

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