Solutions of the Cambridge Senate House Problems And Riders for the Year 1878

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Viii. Prove that the complete solution of a partial differential equation of the first order containing two independent variables is afforded by the system consisting of a single complete primitive with its accompanying general integrals and singular solution. Find a complete primitive of the equation dz (dz\ (dz~Y \dx) \dy) Present the complete integral of the equation 1 d*u _ d?u 2 du n (n + 1) u ~ ~~ . M the form u n /l d\ n = r - ^- \r drj r To find a complete primitive of the equation Cons...ider x and y as the co-ordinates of a point, and turn the axes round through half a right angle ; that is to say, take By this transformation the equation will become 158 SOLUTIONS OF SENATE-HOUSE [WEDNESDAY, to find a complete primitive of which, put where a is an arbitrary constant ; therefore J2 Z = fj@? - *) ^ + (*/(* + rf) dr, + b, b being another arbitrary constant. These integrals belong to known forms, and may be readily found. To present the complete integral of the equation 1 d*u_d*u Idu n(n+\) .

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