I have an equation: $$f''''(x) = f'''(x) \frac{\partial H}{\partial y'''} + f''(x) \frac{\partial H}{\partial y''}.$$ How can I solve this equation for $H$, where $H$ is a function of $x, y'', and \ \ y''' \ \ and \ \ f''$ is non - zero. Any help or hints will be appreciated. Thanks
2026-05-04 11:04:48.1777892688
Diffferential Equations
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Method of characteristics will definitely work. Also the standard Ansatz for transport equations will work and both yield: $$ H(x,y'',y''')=G\Big(y''f'''(x)-y'''f''(x),x\Big)+y''\tfrac{f''''(x)}{f''(x)} $$ For any function $G$ smooth enough. Notice that the variable $x$ is only a parameter here, if you meant something different please clarify in the OP. Similarly, $y''$ and $y'''$ are treated as variables not as functions.