Question

If =2x+3x+d is the general solution for dy/dx +(4x+3)y =0 , determine the particular solution for the equation when y(2) =1.

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Answer 1

`y=e^(-2x^2-3x+14)`

Explanation:

Determine the general solution

`(dy)/(dx)+(4x+3)y=0,\: y(2)=1\\ \\(dy)/(dx)+4xy+3y=0\\ \\(dy)/(dx)=-4xy-3y\\ \\dy=(-4xy-3y)dx\\\\(1)/(y)dy=(-4x-3)dx\\ \\\int {(1)/(y) } \, dy=\int{(-4x-3)} \, dx\\ \\ ln(|y|)=-2x^2-3x+C\\`

Solve for C given the initial condition y(2)=1

`ln(|1|)=-2(2)^2-3(2)+C\\\\0=-2(4)-6+C\\\\0=-8-6+C\\\\0=-14+C\\\\14=C`

Plug the value of C into the general solution equation

`ln(|y|)=-2x^2-3x+14\\\\y=e^(-2x^2-3x+14)`