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Question Number 60357 by Sardor2211 last updated on 20/May/19

Commented by Mr X pcx last updated on 20/May/19

$$theequationisnotclearbutisuppose$$$$thatis(1+x^2)y^{{}^{\prime }}-2xy={(1+x^2)}^2$$$$\left(he\right)\Rightarrow (1+x^2)y^{{}^{\prime }}-2xy=0\Rightarrow$$$$(1+x^2)y^{{}^{\prime }}=2xy\Rightarrow \frac{y^{{}^{\prime }}}{y}=\frac{2x}{1+x^2}\Rightarrow$$$$ln\mid y\mid =ln(1+x^2)+c\Rightarrow$$$$y=k(1+x^2)mvcmrthodgive$$$$y^{{}^{\prime }}=k^{,}(1+x^2)+2xk$$$$\left(e\right)\Rightarrow (1+x^2)k^{\prime }(1+x^2)+2xk(1+x^2)$$$$-2xk(1+x^2)={(1+x^2)}^2\Rightarrow$$$$k^{{}^{\prime }}=1\Rightarrow k\left(x\right)=x+\lambda \Rightarrow$$$$y\left(x\right)=(1+x^2)(x+\lambda )\Rightarrow$$$$y\left(x\right)=x^3+x+\lambda (1+x^2)$$$$weseethatyisapolynom..$$

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