Solve $\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{y}{x}=1$, using the substitution $y=vx$. Solve \(\f

Question

Solve $\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{y}{x}=1$, using the substitution $y=vx$. Solve $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y-x+1}{y-x+5}$, using the substitution $z=y-x$. Solve $2x^{3}\frac{\mathrm{d}y}{\mathrm{d}x}=x^{2}y+1$, using the substitution $y=u\sqrt{x}$. Solve $\frac{\mathrm{d}y}{\mathrm{d}x}+\sin 2y=\sin^{2}y$, using the substitution $z=\tan y$.

Accepted Answer

(i) $y=x\ln|x|+Cx$ (ii) $\dfrac{(y-x)^2}{2}+5(y-x)+4x=C$ (iii) $2y=-\dfrac{2}{5x^2}+C\sqrt x$ (iv) $y=\tan^{-1}\left(\dfrac2{1-Ae^{2x}}\right)+n\pi$, with the additional constant solutions $y=n\pi$

Differential Equations - Solving Differential Equations by Separation of Variables

Differential Equations - Solving Differential Equations by Separation of Variables

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