Question

Calculus - give general solution in implicit form to following

Answer 1

\[dy/dx =(2y ^{1/2} (e ^{2x}-5x))/(3(e ^{2x}-5x)^{2/3})\]

Answer 2

\[\int\limits dy/y^{1/2}=\int\limits (2/3)(e^{2x}-5x)^{1/3}dx\]

Answer 3

\[dy/dx =(2y ^{1/2} (e ^{2x}-5x))/(3(e ^{2x}-5x)^{2/3})\]
\[\int\limits\limits \frac{1}{2\sqrt y}~dy=\int\limits\limits (2/3)(e^{2x}-5x)^{1/3}dx\]
\[\text{}\int \frac{1}{2 \sqrt{y}} \, dy=\frac{1}{3}\int\limits \left(e^{2 x}-5 x\right) \, dx\]
\[\frac{1}{2}\int \frac{1}{\sqrt{y}} \, dy=\frac{1}{3}\int\limits e^{2 x} \, dx-\frac{5}{3}\int\limits x \, dx\]
\[\sqrt{y}=\frac{e^{2 x}}{6}-\frac{5 x^2}{6}\]
\[\sqrt{y}=\frac{1}{6} \left(e^{2 x}-5 x^2\right)\]
\[y=(\frac{1}{6} \left(e^{2 x}-5 x^2\right))^2\]