Question

Answer the following and give complete solutions.

1. Find the constant a so that phi(x)=x(e^2x) is a solution to y'+ay-e^(2x)=0
2. Find the constant a so that e^xy + y = x-1 is an implicit solution to dy/dx = e^-xy +ay / e^-xy + x.

Answer 1

A)Givenϕ (x) = xe^2x

$\phi '= e^{2x}(1 + 2x)$

If  ϕ(x) is a solution to the given equation then it satisfies the equation,

$e^{2x}(1+2x)+axe^{2x}-e^{2x}=0 \\ 1+2x+ax-1=0 \\ 2x+ax=0 \\ a=-2$

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B)

Given y(e^x+1)=x-1

Differentiating with respect to x

dy ( e^x+1) + y(e^x) dx = dx

dy(e^x+1) = dx ( 1-ye^x)

$\frac{dy}{dx} = \frac{1-ye^{x}}{e^{x} +1}$

Putting this value in the given differential equation

$\frac{1-ye^{x}}{e^{x} +1}=e^{-x}y+\frac{ay}{e^{-x}y}+x\\$

$\frac{1-ye^{x}}{e^{x} +1}-e^{-x}y-x$ =$\frac{a}{e^{-x}}$

 

$\frac{1-ye^{-x}-y(1+e^x)-x(1+e^x)}{1+e^x}= ae^x \\ \\ \frac{1-ye^{-x}-(1+e^x)(x+y)}{e^x(1+e^x)}= a$