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4.47 Problems 2301 to 2313

4.47.1 \(2 x y''(x) y'''(x)=y''(x)^2-a^2\)
4.47.2 \(\left (1-x^2\right ) \left (y'''(x)\right )^2+2 x y''(x) y'''(x)-y''(x)^2+1=0\)
4.47.3 \(\sqrt {y''(x)^2+1} \left (1-y'''(x)\right )=y''(x) y'''(x)\)
4.47.4 \(3 y''(x) y''''(x)=5 \left (y'''(x)\right )^2\)
4.47.5 \(-4 y'(x) \left (y'''(x)\right )^2+3 y'(x) y''(x) y''''(x)-3 y''(x)^3=0\)
4.47.6 \(-45 y''(x) y'''(x) y''''(x)+9 y''(x)^2 y'''''(x)+40 \left (y'''(x)\right )^3=0\)
4.47.7 \(y^{(n)}(x)=f(x)\)
4.47.8 \(y^{(n)}(x)=f\left (\genfrac {}{}{}{}{\partial ^{n-2}y(x)}{\partial x^{n-2}}\right )\)
4.47.9 \(y^{(n)}(x)=f\left (\genfrac {}{}{}{}{\partial ^{n-1}y(x)}{\partial x^{n-1}}\right )\)
4.47.10 \(y^{(n)}(x) \genfrac {}{}{}{}{\partial ^{n-2}y(x)}{\partial x^{n-2}}=\left (\genfrac {}{}{}{}{\partial ^{n-1}y(x)}{\partial x^{n-1}}\right )^2\)
4.47.11 \(f\left (x,y^{(n)}(x)\right )=0\)
4.47.12 \(f\left (x,\genfrac {}{}{}{}{\partial ^{n-1}y(x)}{\partial x^{n-1}},y^{(n)}(x)\right )=0\)
4.47.13 \(f\left (\genfrac {}{}{}{}{\partial ^{n-2}y(x)}{\partial x^{n-2}},\genfrac {}{}{}{}{\partial ^{n-1}y(x)}{\partial x^{n-1}},y^{(n)}(x)\right )=0\)

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