The given DE can be written in the form:
y21dxdy+y1=2x which is a Bernoulli type of first order DE.
Solve Bernoulli’s equation y(x)1+y(x)2dxdy(x)=2x:Multiply both sides by −1:−y(x)2dxdy(x)−y(x)1=−2xLet v(x)=y(x)1, which gives dxdv(x)=−y(x)2dxdy(x):dxdv(x)−v(x)=−2x Let μ(x)=e∫−1dx=e−x.Multiply both sides by μ(x):e−xdxdv(x)−e−xv(x)=−2e−xxSubstitute −e−x=dxd(e−x):e−xdxdv(x)+dxd(e−x)v(x)=−2e−xx
Apply the reverse product rule fdxdg+gdxdf=dxd(fg) to the left-hand side:dxd(e−xv(x))=−2e−xxIntegrate both sides with respect to x:∫dxd(e−xv(x))dx=∫−2e−xxdxEvaluate the integrals:e−xv(x)=−2e−x(−x−1)+c1, where c1 is an arbitrary constant.Divide both sides by μ(x)=e−x:v(x)=2x+c1ex+2 Solving for y(x), we have:
y(x)=v(x)1=2x+c1ex+21
Which is the required solution of the given DE.
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