Differential Equations

Find the particular solution of the differential equation - Wired Faculty

Question

Find the particular solution of the differential equation $log open parentheses dy over dx close parentheses equals 3 straight x plus 4 straight y comma space given space that space straight y space equals space 0 comma space when space straight x equals space 0.$

Solution

Consider the differential equation,
$log open parentheses dy over dx close parentheses equals 3 straight x plus 4 straight y$
Taking exponent on both the sides, we have
$straight e to the power of log open parentheses dy over dx close parentheses end exponent space equals space straight e to the power of 3 straight x plus 4 straight y end exponent space rightwards double arrow dy over dx equals straight e to the power of 3 straight x plus 4 straight y end exponent space rightwards double arrow dy over dx equals straight e to the power of 3 straight x end exponent. straight e to the power of 4 straight y end exponent rightwards double arrow dy over straight e to the power of 4 straight y end exponent equals straight e to the power of 3 straight x end exponent. dx$
Integration in both the sides, we have
$integral dy over straight e to the power of 4 straight y end exponent space equals space integral straight e to the power of 3 straight x end exponent dx fraction numerator straight e to the power of negative 4 straight y end exponent over denominator negative 4 end fraction equals straight e to the power of 3 straight x end exponent over 3 plus straight C We space need space to space find space the space particular space solution We space have comma space straight y space equals space 0 comma space when space straight x space equals space 0 fraction numerator 1 over denominator negative 4 end fraction space equals 1 third plus straight C rightwards double arrow straight C space equals negative 1 fourth minus 1 third rightwards double arrow straight C space equals space fraction numerator negative 3 minus 4 over denominator 12 end fraction equals negative 7 over 12 Thus comma space the space solution space is space straight e to the power of 3 straight x end exponent over 3 plus straight e to the power of negative 4 straight y end exponent over 4 equals 7 over 12$

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Differential Equations

Find the particular solution of the differential equation $log open parentheses dy over dx close parentheses equals 3 straight x plus 4 straight y comma space given space that space straight y space equals space 0 comma space when space straight x equals space 0.$

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