Show that the differential equation $y\left( \frac{\mathrm{d}y}{\mathrm{d}x}+2y \right)=\frac{x}{{{\

Question

Show that the differential equation $y\left( \frac{\mathrm{d}y}{\mathrm{d}x}+2y \right)=\frac{x}{{{\mathrm{e}}^{4x+{{x}^{2}}}}}$ can be reduced by the substitution $z={{y}^{2}}{{\mathrm{e}}^{4x}}$ to $\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{2x}{{{\mathrm{e}}^{{{x}^{2}}}}}$. Hence, find the general solution in the form ${{y}^{2}}=\mathrm{f}\left( x \right)$. [4]

Accepted Answer

$\dfrac{\mathrm dz}{\mathrm dx}=\dfrac{2x}{e^{x^2}}$ $y^2=Ce^{-4x}-e^{-x^2-4x}$

2025 YIJC P1 Q1

2025 YIJC P1 Q1

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