Solve the linear ODE y 2y e2xSolutionGiven that y 2y e2x

Solve the linear O.D.E.: y\' + 2y = e^-2x

Solution

Given that

y\' + 2y = e^-2x

The given single order linear equation is in the form of y\' + p(x)y = q(x) then the general solution is ,

   y.e ^p(x)dx =   q(x).e ^p(x)dx + c.................................1

y\' + 2y = e^-2x

Hence,

p(x) = 2 , q(x) =   e^-2x

e ^p(x)dx = e ^{2 dx}

   =e ^{2. dx}

   = e^2x [since, dx = x ]

Substitute e ^p(x)dx = e^2x , q(x) = e^-2x in equation 1

   y.e ^p(x)dx =   q(x).e ^p(x)dx + c

y.e^2x =   e^-2x.e^2x + c

   ye^2x = e^{-2x + 2x} + c [ since, a^m.aⁿ = a^m+n ]

=   e⁰dx + c

   = 1 dx + c [ since, e⁰ = 1 ]

= 1. dx + c

= 1.x + c

ye^2x = x + c

y = ( x + c ) / e^2x

y = e^-2x( x + c ) [ since, 1/a^m = a^-m ]

Therefore,

The general solution is , y = e^-2x( x + c )