$2 \frac{d t}{d x}=\frac{1}{x\left(x^{2} \sin t+1\right)}$
$\frac{d x}{d t}=\frac{x^{3} \sin t}{2}+\frac{x}{2}$
$x^{-3} \frac{d x}{d t}=\frac{\sin t}{2}+\frac{x^{-2}}{2}$
Let $x^{-2}=z \Rightarrow x^{-3} \frac{d x}{d t}=-\frac{d z}{2 d t}$
$-\frac{1}{2} \frac{d z}{d t}=\frac{\sin t}{2}+\frac{z}{2}$
$\frac{d z}{d t}=-\sin t-z$
$\frac{d z}{d t}+z=-\sin t$
$\mathrm{ze}^{\mathrm{t}}=-\int \mathrm{e}^{\mathrm{t}} \sin \mathrm{t} d \mathrm{t}$
$\mathrm{ze}^{\mathrm{t}}=\frac{\left(-\mathrm{e}^{\mathrm{t}} \cos \mathrm{t}+\mathrm{e}^{\mathrm{t}} \sin \mathrm{t}\right)}{2}+\mathrm{C}$
$\frac{e^{y^{2}}}{x^{2}}=e^{t} \frac{(\cos t-\sin t)}{2}+C$
$e^{y^{2}}\left(\frac{1}{x^{2}}-\frac{\cos t}{2}+\frac{\sin t}{2}\right) \pm C$
$e^{y^{2}}\left(\frac{1}{x^{2}}-\frac{\cos y^{2}}{2}+\frac{\sin y^{2}}{2}\right)=C$
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વિધાન $2$: કોઇપણ શ્રેણિક $A$ માટે $\det \left( {{A^T}} \right) = {\rm{det}}\left( A \right)$ અને $\det \left( { - A} \right) = - {\rm{det}}\left( A \right)$ જયાં $\det \left( A \right) = A$ નો નિશ્રાયક.