Word Count : 253

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मान लेते हैं कि समीकरण का अर्थ है $$ \frac{dy}{dx}=x\,y^{4}+\frac{x^{4}}{x\,y^{3}}=x\,y^{4}+x^{3}y^{-3}. $$ पहला कदम समीकरण को सरल बनाने के लिए उपयुक्त प्रतिलोम परिवर्तन लेना है। प्रयुक्त परिवर्तन लीजिए $$ u=y^{4}\quad\Rightarrow\quad \frac{du}{dx}=4y^{3}\frac{dy}{dx}. $$ अब मूल समीकरण में $\dfrac{dy}{dx}$ की जगह रखकर: $$ \frac{du}{dx}=4y^{3}\big(xy^{4}+x^{3}y^{-3}\big)=4x\,y^{7}+4x^{3} . $$ चूँकि $y^{7}=y^{4}\,y^{3}=u\,u^{3/4}=u^{7/4}$, हमें मिलता है $$ \boxed{\ __________ ____ _______ _____ _________.
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