Question

Solve the given initial value problem.

dy/dx = y(2x^2 +y^2)/2x^3

subject to y(1) = 1.

Answer 1

$\text{ Solve Bernoulli's equation } \frac{d y(x)}{d x}=\frac{1}{2} x^{3}\left(2 x^{2}+y(x)^{2}\right) y(x), \text{ such that } y(1)=1 :\\[4mm] \text{ Rewrite the equation:}\\[2mm] \frac{d y(x)}{d x}-x^{5} y(x)=\frac{1}{2} x^{3} y(x)^{3}\\[4mm] \text{ Divide both sides by } -\frac{1}{2} y(x)^{3} :\\[2mm] -\frac{2 \frac{d y(x)}{d x}}{y(x)^{3}}+\frac{2 x^{5}}{y(x)^{2}}=-x^{3}\\[4mm] \text{ Let } v(x)=\frac{1}{y(x)^{2}},\text{ which gives } \frac{d v(x)}{d x}=-\frac{2 \frac{d y(x)}{d x}}{y(x)^{3}} :\\[2mm] \frac{d v(x)}{d x}+2 x^{5} v(x)=-x^{3}\\[3mm] \text{ Let } \mu(x)=e^{\int 2 x^{5} d x}=e^{x^{6} / 3}.\\[4mm] \text{ Multiply both sides by }\mu(x) :\\[2mm] e^{x^{6} / 3} \frac{d v(x)}{d x}+\left(2 e^{x^{6} / 3} x^{5}\right) v(x)=-e^{x^{6} / 3} x^{3} \text{ Substitute } 2 e^{x^{6} / 3} x^{5}=\frac{d}{d x}\left(e^{x^{6} / 3}\right):\\[2mm] e^{x^{6} / 3} \frac{d v(x)}{d x}+\frac{d}{d x}\left(e^{x^{6} / 3}\right) v(x)=-e^{x^{6} / 3} x^{3}$

$\text{ Apply the reverse product rule } f \frac{d g}{d x}+g \frac{d f}{d x}=\frac{d}{d x}(f g) \text{ to the left-hand side: } \frac{d}{d x}\left(e^{x^{6} / 3} v(x)\right)=-e^{x^{6} / 3} x^{3}\\[4mm] \text{ Integrate both sides with respect to } x :\\[2mm] \int \frac{d}{d x}\left(e^{x^{6} / 3} v(x)\right) d x=\int-e^{x^{6} / 3} x^{3} d x\\[4mm] \text{Evaluate the integrals:}\\[2mm] e^{x^{6} / 3} v(x)=-\frac{\sqrt[3]{-x^{6}} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3} x^{2}}+c_{1}, \text{ where } c_{1} \text{ is an arbitrary constant.}\\[4mm] \text{ Divide both sides by }\mu(x)=e^{x^{6} / 3}:\\[2mm] v(x)=e^{-x^{6} / 3}\left(\frac{x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3}\left(-x^{6}\right)^{2 / 3}}+c_{1}\right)\\[4mm] \text{ Solve for }y(x) \text{ in } v(x)=\frac{1}{y(x)^{2}}\\[2mm] y(x)=-\frac{e^{x^{6} / 6} \sqrt[3]{-x^{6}}}{\sqrt{\frac{x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3}}+c_{1}\left(-x^{6}\right)^{2 / 3}}} \text { or } y(x)=\frac{e^{x^{6} / 6} \sqrt[3]{-x^{6}}}{\sqrt{\frac{x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3}}+c_{1}\left(-x^{6}\right)^{2 / 3}}}$

$\text{ For } y(x)=-\frac{e^{x^{6} / 6} \sqrt[3]{-x^{6}}}{\sqrt{\frac{x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3}}+\left(-x^{6}\right)^{2 / 3} c_{1}}}, \text{ solve for } c_{1} \text{ using the initial conditions:}\\[4mm] \text{ Substitute }y(1)=1 \text{ into } y(x)=-\frac{e^{x^{6} / 6 \sqrt[3]{-x^{6}}}}{\sqrt{\frac{x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3}}+\left(-x^{6}\right)^{2 / 3} c_{1}}} :\\[2mm] -\frac{\sqrt[3]{-1} \sqrt[6]{e}}{\sqrt{(-1)^{2 / 3} c_{1}+\frac{r\left(\frac{2}{3},-\frac{1}{3}\right)}{2 \sqrt[3]{3}}}}=1\\[4mm] \text{ The equation has no solution.}\\[2mm] y(x)=-\frac{e^{x^{6} / 6} \sqrt[3]{-x^{6}}}{\sqrt{\frac{x^{4} r\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3}}+c_{1}\left(-x^{6}\right)^{2 / 3}}}\\[2mm] \text{ cannot satisfy the initial condition, which means no solution exists.}\\[2mm] \text{ For } y(x)=\frac{e^{x^{6} / 6} \sqrt[3]{-x^{6}}}{\sqrt{\frac{x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3}}+\left(-x^{6}\right)^{2 / 3} c_{1}}}, \text{ solve for } c_{1} \text{ using the initial conditions:}\\[4mm] \text{ Substitute } y(1)=1 \text{ into } y(x)=\frac{e^{x^{6} / 6} \sqrt[3]{-x^{6}}}{\sqrt{\frac{x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3}}+\left(-x^{6}\right)^{2 / 3} c_{1}}}:$

$\frac{\sqrt[3]{-1} \sqrt[6]{e}}{\sqrt{(-1)^{2 / 3} c_{1}+\frac{\Gamma\left(\frac{2}{3},-\frac{1}{3}\right)}{2 \sqrt[3]{3}}}}=1\\[4mm] \text{Solve the equation:}\\[2mm] c_{1}=\frac{1}{6}\left(\frac{1}{2}\left(12 \sqrt[3]{e}+3^{2 / 3} \Gamma\left(\frac{2}{3},-\frac{1}{3}\right)\right)+\frac{3}{2} i \sqrt[6]{3} \Gamma\left(\frac{2}{3},-\frac{1}{3}\right)\right)\\[4mm] \text{ Substitute } c_{1}=\frac{1}{6}\left(\frac{1}{2}\left(12 \sqrt[3]{e}+3^{2 / 3} \Gamma\left(\frac{2}{3},-\frac{1}{3}\right)\right)+\frac{3}{2} i \sqrt[6]{3} \Gamma\left(\frac{2}{3},-\frac{1}{3}\right)\right) \text{ into } y(x)=\frac{e^{x^{6} / 6} \sqrt[3]{-x^{6}}}{\sqrt{\frac{x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)}{2 \sqrt[3]{3}}+\left(-x^{6}\right)^{2 / 3} c_{1}}}\\[2mm] y(x)=\frac{2 \sqrt{3} e^{x^{6} / 6 \sqrt[3]{-x^{6}}}}{\sqrt{2 \times 3^{2 / 3} x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)+\left(\sqrt[6]{3} \Gamma\left(\frac{2}{3},-\frac{1}{3}\right)(3 i+\sqrt{3})+12 \sqrt[3]{e}\right)\left(-x^{6}\right)^{2 / 3}}}\\[4mm] \text{ Collecting the solutions, we have:} \\[2mm] y(x) = \frac{2 \sqrt{3} e^{x^{6} / 6} \sqrt[3]{-x^{6}}}{\sqrt{2 \times 3^{2 / 3} x^{4} \Gamma\left(\frac{2}{3},-\frac{x^{6}}{3}\right)+\left(\sqrt[6]{3} \Gamma\left(\frac{2}{3},-\frac{1}{3}\right)(3 i+\sqrt{3})+12 \sqrt[3]{e}\right)\left(-x^{6}\right)^{2 / 3}}}$