09. Mixed Examples

Dated: 10-11-2024

Example

\[y^\prime=\frac{x^2+y^2}{xy}\]

\[\frac{dy}{dx}=\frac{x^2+y^2}{xy}\]

Put \(y = wx\) then

\[\frac{dy}{dx}=w+x\frac{dw}{dx}\]

\[w+x\frac{dw}{dx}=\frac{x^2+w^2x^2}{x\times w}=\frac{1+w^2}{w}\]

\[w+x\frac{dw}{dx}=\frac{1}{w}+w\]

\[wdw=\frac{dx}{x}\]

After integration¹

\[\frac{w^2}{2}=\ln x+\ln c\]

\[\frac{y^2}{2x^2}=\ln|xc|\]

\[y^2=2x^2\ln|xc|\]

\[xe^{2y}\frac{dy}{dx}+e^{2y}=\frac{\ln x}{x}\]

Put \(e^{2y} = u\) and we get

\[2e^{2y}\frac{dy}{dx}=\frac{du}{dx}\]

\[\frac{x}{2}\frac{du}{dx}+u=\frac{\ln x}{x}\]

\[\frac{du}{dx}+\frac{2}{x}u=2\frac{\ln x}{x^{2}}\]

\[u(x) =\exp\left(\int\frac{2}{x}dx\right)=x^{2}\]

\[x^{2}\frac{du}{dx}+2xu=2\ln x\]

\[\frac{d}{dx}(x^{2}u)=2\ln x\]

\[x^{2}u=2[x \ln x-x]+c\]

\[x^{2}e^{2y}=2[x \ln x-x]+c\]

\[x^4y^2y'+x^3y^3=2x^3-3\]

Put \(x^3y^3 = u\) and we get

\[3x^{2}y^{3}+3x^{3}y^{2}\frac{dy}{dx}=\frac{du}{dx}\]

\[3x^{3}y^{2}\frac{dy}{dx}=\frac{du}{dx}-3x^{2}y^{3}\]

\[x^{4}y^{2}\frac{dy}{dx}=\frac{x}{3}\frac{du}{dx}-x^{3}y^{3}\]

\[\frac{x}{3}\frac{du}{dx}=2x^{3}-3\]

\[\frac{du}{dx}=6x^{2}-\frac{9}{x}\]

Integrating¹ both sides, we get

\[u=2x^3-9\ln x+c\]

\[x^3y^3=2x^3-9\ln x+c\]