Question

Solve the given differential equation by using an appropriate substitution. The DE is of the form dy dx = f(Ax + By + C), which is given in (5) of Section 2.5. dy dx = 4 + y − 4x + 5

Answer 1

No idea what the cited section's method is, but this ODE is linear:

`(\mathrm dy)/(\mathrm dx)=4+y-4x+5`

`(\mathrm dy)/(\mathrm dx)-y=9-4x`

Multiply both sides by
`e^(-x)` so that the left side can be condensed as the derivative of a product:

`e^(-x)(\mathrm dy)/(\mathrm dx)-e^(-x)y=(9-4x)e^(-x)`

`(\mathrm d)/(\mathrm dx)\left[e^(-x)y\right]=(9-4x)e^(-x)`

Integrating both sides gives

`e^(-x)y=(4x-5)e^(-x)+C`

`\implies\boxed{y(x)=4x-5+Ce^x}`