ODE5.2

From Example Problems
Jump to: navigation, search

y'+xy=xy^{2}\,

A Bernoulli Equation has the form

y'+p(x)y=q(x)y^{n}\,

And for these types of DE's a substitution of z=y^{{1-n}}\, is recommended. In the present problem, n=2\, and so

z=y^{{-1}}\,

y=z^{{-1}}\,

y'=-{\frac  {z'}{z^{2}}}\,

Substituting these new variables into the original equation,

-{\frac  {z'}{z^{2}}}+{\frac  {x}{z}}={\frac  {x}{z^{2}}}\,

z'-xz=-x\,

Use the integrating factor e^{{\int -xdx}}=e^{{-x^{2}/2}}\,

e^{{-x^{2}/2}}z'-e^{{-x^{2}/2}}xz=-e^{{-x^{2}/2}}x\,

{\frac  {d}{dx}}[e^{{-x^{2}/2}}z]=-e^{{-x^{2}/2}}x\,

e^{{-x^{2}/2}}z=e^{{-x^{2}/2}}+c_{1}\,

z=1+c_{1}e^{{x^{2}/2}}\,

Undoing the substitution z=y^{{-1}},

y^{{-1}}=1+c_{1}e^{{x^{2}/2}}\,

y=(1+c_{1}e^{{x^{2}/2}})^{{-1}}\,

Ordinary Differential Equations