Question #292775

Solve the first order linear inhomogeneous differential equation using the bernoulli method

xy,-2y=2x4

1
Expert's answer
2022-02-06T12:04:32-0500

y2yx=2x3y' - 2\frac{y}{x} = 2{x^3}

Substitution: y=uvy=uv+uvy = uv \Rightarrow y' = u'v + uv'

Then

uv+uv2uvx=2x3u'v + uv' -2 \frac{{uv}}{x} = 2{x^3}

uv+u(v2vx)=2x3u'v + u\left( {v' - 2\frac{v}{x}} \right) = 2{x^3}

Let

v2vx=0dvdx=2vxdvv=2dxxlnv=lnx2v=x2v' - \frac{{2v}}{x} = 0 \Rightarrow \frac{{dv}}{{dx}} = \frac{{2v}}{x} \Rightarrow \frac{{dv}}{v} = \frac{{2dx}}{x} \Rightarrow \ln v = \ln {x^2} \Rightarrow v = {x^2}

Then

ux2=2x3u=2xu=x2+Cy=uv=(x2+C)x2=x4+Cx2u'{x^2} = 2{x^3} \Rightarrow u' = 2x \Rightarrow u = {x^2} + C \Rightarrow y = uv = \left( {{x^2} + C} \right){x^2} = {x^4} + C{x^2}

Answer: y=x4+Cx2y = {x^4} + C{x^2}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS