Solve Bernoulli’s equation d y ( x ) d x = 1 2 x 3 ( 2 x 2 + y ( x ) 2 ) y ( x ) , such that y ( 1 ) = 1 : Rewrite the equation: d y ( x ) d x − x 5 y ( x ) = 1 2 x 3 y ( x ) 3 Divide both sides by − 1 2 y ( x ) 3 : − 2 d y ( x ) d x y ( x ) 3 + 2 x 5 y ( x ) 2 = − x 3 Let v ( x ) = 1 y ( x ) 2 , which gives d v ( x ) d x = − 2 d y ( x ) d x y ( x ) 3 : d v ( x ) d x + 2 x 5 v ( x ) = − x 3 Let μ ( x ) = e ∫ 2 x 5 d x = e x 6 / 3 . Multiply both sides by μ ( x ) : e x 6 / 3 d v ( x ) d x + ( 2 e x 6 / 3 x 5 ) v ( x ) = − e x 6 / 3 x 3 Substitute 2 e x 6 / 3 x 5 = d d x ( e x 6 / 3 ) : e x 6 / 3 d v ( x ) d x + d d x ( e x 6 / 3 ) v ( x ) = − e x 6 / 3 x 3 \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} Solve Bernoulli’s equation d x d y ( x ) = 2 1 x 3 ( 2 x 2 + y ( x ) 2 ) y ( x ) , such that y ( 1 ) = 1 : Rewrite the equation: d x d y ( x ) − x 5 y ( x ) = 2 1 x 3 y ( x ) 3 Divide both sides by − 2 1 y ( x ) 3 : − y ( x ) 3 2 d x d y ( x ) + y ( x ) 2 2 x 5 = − x 3 Let v ( x ) = y ( x ) 2 1 , which gives d x d v ( x ) = − y ( x ) 3 2 d x d y ( x ) : d x d v ( x ) + 2 x 5 v ( x ) = − x 3 Let μ ( x ) = e ∫ 2 x 5 d x = e x 6 /3 . Multiply both sides by μ ( x ) : e x 6 /3 d x d v ( x ) + ( 2 e x 6 /3 x 5 ) v ( x ) = − e x 6 /3 x 3 Substitute 2 e x 6 /3 x 5 = d x d ( e x 6 /3 ) : e x 6 /3 d x d v ( x ) + d x d ( e x 6 /3 ) v ( x ) = − e x 6 /3 x 3
Apply the reverse product rule f d g d x + g d f d x = d d x ( f g ) to the left-hand side: d d x ( e x 6 / 3 v ( x ) ) = − e x 6 / 3 x 3 Integrate both sides with respect to x : ∫ d d x ( e x 6 / 3 v ( x ) ) d x = ∫ − e x 6 / 3 x 3 d x Evaluate the integrals: e x 6 / 3 v ( x ) = − − x 6 3 Γ ( 2 3 , − x 6 3 ) 2 3 3 x 2 + c 1 , where c 1 is an arbitrary constant. Divide both sides by μ ( x ) = e x 6 / 3 : v ( x ) = e − x 6 / 3 ( x 4 Γ ( 2 3 , − x 6 3 ) 2 3 3 ( − x 6 ) 2 / 3 + c 1 ) Solve for y ( x ) in v ( x ) = 1 y ( x ) 2 y ( x ) = − e x 6 / 6 − x 6 3 x 4 Γ ( 2 3 , − x 6 3 ) 2 3 3 + c 1 ( − x 6 ) 2 / 3 or y ( x ) = e x 6 / 6 − x 6 3 x 4 Γ ( 2 3 , − x 6 3 ) 2 3 3 + c 1 ( − x 6 ) 2 / 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}}} Apply the reverse product rule f d x d g + g d x df = d x d ( f g ) to the left-hand side: d x d ( e x 6 /3 v ( x ) ) = − e x 6 /3 x 3 Integrate both sides with respect to x : ∫ d x d ( e x 6 /3 v ( x ) ) d x = ∫ − e x 6 /3 x 3 d x Evaluate the integrals: e x 6 /3 v ( x ) = − 2 3 3 x 2 3 − x 6 Γ ( 3 2 , − 3 x 6 ) + c 1 , where c 1 is an arbitrary constant. Divide both sides by μ ( x ) = e x 6 /3 : v ( x ) = e − x 6 /3 ( 2 3 3 ( − x 6 ) 2/3 x 4 Γ ( 3 2 , − 3 x 6 ) + c 1 ) Solve for y ( x ) in v ( x ) = y ( x ) 2 1 y ( x ) = − 2 3 3 x 4 Γ ( 3 2 , − 3 x 6 ) + c 1 ( − x 6 ) 2/3 e x 6 /6 3 − x 6 or y ( x ) = 2 3 3 x 4 Γ ( 3 2 , − 3 x 6 ) + c 1 ( − x 6 ) 2/3 e x 6 /6 3 − x 6
For y ( x ) = − e x 6 / 6 − x 6 3 x 4 Γ ( 2 3 , − x 6 3 ) 2 3 3 + ( − x 6 ) 2 / 3 c 1 , solve for c 1 using the initial conditions: Substitute y ( 1 ) = 1 into y ( x ) = − e x 6 / 6 − x 6 3 x 4 Γ ( 2 3 , − x 6 3 ) 2 3 3 + ( − x 6 ) 2 / 3 c 1 : − − 1 3 e 6 ( − 1 ) 2 / 3 c 1 + r ( 2 3 , − 1 3 ) 2 3 3 = 1 The equation has no solution. y ( x ) = − e x 6 / 6 − x 6 3 x 4 r ( 2 3 , − x 6 3 ) 2 3 3 + c 1 ( − x 6 ) 2 / 3 cannot satisfy the initial condition, which means no solution exists. For y ( x ) = e x 6 / 6 − x 6 3 x 4 Γ ( 2 3 , − x 6 3 ) 2 3 3 + ( − x 6 ) 2 / 3 c 1 , solve for c 1 using the initial conditions: Substitute y ( 1 ) = 1 into y ( x ) = e x 6 / 6 − x 6 3 x 4 Γ ( 2 3 , − x 6 3 ) 2 3 3 + ( − x 6 ) 2 / 3 c 1 : \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}}}: For y ( x ) = − 2 3 3 x 4 Γ ( 3 2 , − 3 x 6 ) + ( − x 6 ) 2/3 c 1 e x 6 /6 3 − x 6 , solve for c 1 using the initial conditions: Substitute y ( 1 ) = 1 into y ( x ) = − 2 3 3 x 4 Γ ( 3 2 , − 3 x 6 ) + ( − x 6 ) 2/3 c 1 e x 6 /6 3 − x 6 : − ( − 1 ) 2/3 c 1 + 2 3 3 r ( 3 2 , − 3 1 ) 3 − 1 6 e = 1 The equation has no solution. y ( x ) = − 2 3 3 x 4 r ( 3 2 , − 3 x 6 ) + c 1 ( − x 6 ) 2/3 e x 6 /6 3 − x 6 cannot satisfy the initial condition, which means no solution exists. For y ( x ) = 2 3 3 x 4 Γ ( 3 2 , − 3 x 6 ) + ( − x 6 ) 2/3 c 1 e x 6 /6 3 − x 6 , solve for c 1 using the initial conditions: Substitute y ( 1 ) = 1 into y ( x ) = 2 3 3 x 4 Γ ( 3 2 , − 3 x 6 ) + ( − x 6 ) 2/3 c 1 e x 6 /6 3 − x 6 :
− 1 3 e 6 ( − 1 ) 2 / 3 c 1 + Γ ( 2 3 , − 1 3 ) 2 3 3 = 1 Solve the equation: c 1 = 1 6 ( 1 2 ( 12 e 3 + 3 2 / 3 Γ ( 2 3 , − 1 3 ) ) + 3 2 i 3 6 Γ ( 2 3 , − 1 3 ) ) Substitute c 1 = 1 6 ( 1 2 ( 12 e 3 + 3 2 / 3 Γ ( 2 3 , − 1 3 ) ) + 3 2 i 3 6 Γ ( 2 3 , − 1 3 ) ) into y ( x ) = e x 6 / 6 − x 6 3 x 4 Γ ( 2 3 , − x 6 3 ) 2 3 3 + ( − x 6 ) 2 / 3 c 1 y ( x ) = 2 3 e x 6 / 6 − x 6 3 2 × 3 2 / 3 x 4 Γ ( 2 3 , − x 6 3 ) + ( 3 6 Γ ( 2 3 , − 1 3 ) ( 3 i + 3 ) + 12 e 3 ) ( − x 6 ) 2 / 3 Collecting the solutions, we have: y ( x ) = 2 3 e x 6 / 6 − x 6 3 2 × 3 2 / 3 x 4 Γ ( 2 3 , − x 6 3 ) + ( 3 6 Γ ( 2 3 , − 1 3 ) ( 3 i + 3 ) + 12 e 3 ) ( − x 6 ) 2 / 3 \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}}} ( − 1 ) 2/3 c 1 + 2 3 3 Γ ( 3 2 , − 3 1 ) 3 − 1 6 e = 1 Solve the equation: c 1 = 6 1 ( 2 1 ( 12 3 e + 3 2/3 Γ ( 3 2 , − 3 1 ) ) + 2 3 i 6 3 Γ ( 3 2 , − 3 1 ) ) Substitute c 1 = 6 1 ( 2 1 ( 12 3 e + 3 2/3 Γ ( 3 2 , − 3 1 ) ) + 2 3 i 6 3 Γ ( 3 2 , − 3 1 ) ) into y ( x ) = 2 3 3 x 4 Γ ( 3 2 , − 3 x 6 ) + ( − x 6 ) 2/3 c 1 e x 6 /6 3 − x 6 y ( x ) = 2 × 3 2/3 x 4 Γ ( 3 2 , − 3 x 6 ) + ( 6 3 Γ ( 3 2 , − 3 1 ) ( 3 i + 3 ) + 12 3 e ) ( − x 6 ) 2/3 2 3 e x 6 /6 3 − x 6 Collecting the solutions, we have: y ( x ) = 2 × 3 2/3 x 4 Γ ( 3 2 , − 3 x 6 ) + ( 6 3 Γ ( 3 2 , − 3 1 ) ( 3 i + 3 ) + 12 3 e ) ( − x 6 ) 2/3 2 3 e x 6 /6 3 − x 6
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