2.1.51 Problem 51

Internal problem ID [9122]
Book : Second order enumerated odes
Section : section 1
Problem number : 51
Date solved : Wednesday, March 05, 2025 at 07:26:17 AM
CAS classification : [[_2nd_order, _missing_x]]

Solve

$$\begin{array}{rl}y{y^{\prime \prime }}^3+y^3{y}^{\prime }& =0\end{array}$$

Factoring the ode gives these factors

$$\begin{array}{}\text{(1)}& y& =0\text{(2)}& {y^{\prime \prime }}^3+y^2{y}^{\prime }& =0\end{array}$$

Now each of the above equations is solved in turn.

Solving equation (1)

Solving for $y$ from

$$\begin{array}{r}y=0\end{array}$$

Solving gives $y=0$

Solving equation (2)

Solved as second order missing x ode

Time used: 33.930 (sec)

This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable $y$ an independent variable. Using

$$\begin{array}{rl}{y}^{\prime }& =p\end{array}$$

Then

$$\begin{array}{rl}{y}^{″}& =\frac{dp}{dx}\\ & =\frac{dp}{dy}\frac{dy}{dx}\\ & =p\frac{dp}{dy}\end{array}$$

Hence the ode becomes

$$\begin{array}{r}yp{\left(y\right)}^3{\left(\frac{d}{dy}p\left(y\right)\right)}^3+y^{3}p\left(y\right)=0\end{array}$$

Which is now solved as first order ode for $p(y)$.

Factoring the ode gives these factors

$$\begin{array}{}\text{(1)}& p& =0\text{(2)}& {p^{\prime }}^3{p}^2+y^2& =0\end{array}$$

Now each of the above equations is solved in turn.

Solving equation (1)

Solving for $p$ from

$$\begin{array}{r}p=0\end{array}$$

Solving gives $p=0$

Solving equation (2)

Let $p=p^{\prime }$ the ode becomes

$$\begin{array}{r}{p}^3{p}^2+y^2=0\end{array}$$

Solving for $p$ from the above results in

$$\begin{array}{}\text{(1)}& p& =-\frac{y}{\sqrt{-p}p}\text{(2)}& p& =\frac{y}{\sqrt{-p}p}\end{array}$$

This has the form

$$\begin{array}{}\text{(*)}& p=yf(p)+g(p)\end{array}$$

Where $f,g$ are functions of $p=p^{\prime }(y)$. Each of the above ode’s is dAlembert ode which is now solved.

Solving ode 1A

Taking derivative of (*) w.r.t. $y$ gives

$$\begin{array}{rl}p& =f+(y{f}^{\prime }+g^{\prime })\frac{dp}{dy}\\ \text{(2)}& p-f& =(y{f}^{\prime }+g^{\prime })\frac{dp}{dy}\end{array}$$

Comparing the form $p=yf+g$ to (1A) shows that

$$\begin{array}{rl}f& =\frac{1}{{(-p)}^{3/2}}\\ g& =0\end{array}$$

Hence (2) becomes

$$\begin{array}{}\text{(2A)}& p-\frac{1}{{(-p)}^{3/2}}=\frac{3y{p}^{\prime }\left(y\right)}{2{(-p)}^{5/2}}\end{array}$$

The singular solution is found by setting $\frac{dp}{dy}=0$ in the above which gives

$$\begin{array}{r}p-\frac{1}{{(-p)}^{3/2}}=0\end{array}$$

No valid singular solutions found.

The general solution is found when $\frac{\mathrm{d}p}{\mathrm{d}y}\ne 0$. From eq. (2A). This results in

$$\begin{array}{}\text{(3)}& p^{\prime }\left(y\right)=\frac{2(p\left(y\right)-\frac{1}{{(-p\left(y\right))}^{3/2}}){(-p\left(y\right))}^{5/2}}{3y}\end{array}$$

This ODE is now solved for $p\left(y\right)$. No inversion is needed.

The ode

$$\begin{array}{}\text{(1)}& p^{\prime }\left(y\right)=\frac{2({(-p\left(y\right))}^{5/2}+1)p\left(y\right)}{3y}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}{p}^{\prime }\left(y\right)& =\frac{2({(-p\left(y\right))}^{5/2}+1)p\left(y\right)}{3y}\\ & =f(y)g(p)\end{array}$$

Where

$$\begin{array}{rl}f(y)& =\frac{2}{3y}\\ g(p)& =({(-p)}^{5/2}+1)p\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(p)}dp& =\int f(y)dy\\ \int \frac{1}{({(-p)}^{5/2}+1)p}dp& =\int \frac{2}{3y}dy\end{array}$$

$$-\frac{2\ln (p{\left(y\right)}^2-{(-p\left(y\right))}^{3/2}-p\left(y\right)-\sqrt{-p\left(y\right)}+1)}{5}-\frac{2\ln (\sqrt{-p\left(y\right)}+1)}{5}+\ln (-p\left(y\right))=\ln \left(y^{2/3}\right)+c_1$$

Substituing the above solution for $p$ in (2A) gives

$$\begin{array}{rl}p& =\frac{y}{{\left({({\mathrm{RootOf}(-1+(y^5{\mathrm{e}}^{\frac{15c_1}{2}}+1){\mathit{\text{\_Z}}}^{75}+(-15y^5{\mathrm{e}}^{\frac{15c_1}{2}}-15){\mathit{\text{\_Z}}}^{70}+(105y^5{\mathrm{e}}^{\frac{15c_1}{2}}+105){\mathit{\text{\_Z}}}^{65}+(-455y^5{\mathrm{e}}^{\frac{15c_1}{2}}-455){\mathit{\text{\_Z}}}^{60}+(1365y^5{\mathrm{e}}^{\frac{15c_1}{2}}+1365){\mathit{\text{\_Z}}}^{55}+(-3000y^5{\mathrm{e}}^{\frac{15c_1}{2}}-3003){\mathit{\text{\_Z}}}^{50}+(4975y^5{\mathrm{e}}^{\frac{15c_1}{2}}+5005){\mathit{\text{\_Z}}}^{45}+(-6300y^5{\mathrm{e}}^{\frac{15c_1}{2}}-6435){\mathit{\text{\_Z}}}^{40}+(6075y^5{\mathrm{e}}^{\frac{15c_1}{2}}+6435){\mathit{\text{\_Z}}}^{35}+(-4375y^5{\mathrm{e}}^{\frac{15c_1}{2}}-5005){\mathit{\text{\_Z}}}^{30}+(2250y^5{\mathrm{e}}^{\frac{15c_1}{2}}+3003){\mathit{\text{\_Z}}}^{25}+(-750y^5{\mathrm{e}}^{\frac{15c_1}{2}}-1365){\mathit{\text{\_Z}}}^{20}+(125y^5{\mathrm{e}}^{\frac{15c_1}{2}}+455){\mathit{\text{\_Z}}}^{15}-105{\mathit{\text{\_Z}}}^{10}+15{\mathit{\text{\_Z}}}^5)}^5-1)}^2\right)}^{3/2}}\end{array}$$

Solving ode 2A

Taking derivative of (*) w.r.t. $y$ gives

$$\begin{array}{rl}p& =f+(y{f}^{\prime }+g^{\prime })\frac{dp}{dy}\\ \text{(2)}& p-f& =(y{f}^{\prime }+g^{\prime })\frac{dp}{dy}\end{array}$$

Comparing the form $p=yf+g$ to (1A) shows that

$$\begin{array}{rl}f& =-\frac{1}{{(-p)}^{3/2}}\\ g& =0\end{array}$$

Hence (2) becomes

$$\begin{array}{}\text{(2A)}& p+\frac{1}{{(-p)}^{3/2}}=-\frac{3y{p}^{\prime }\left(y\right)}{2{(-p)}^{5/2}}\end{array}$$

The singular solution is found by setting $\frac{dp}{dy}=0$ in the above which gives

$$\begin{array}{r}p+\frac{1}{{(-p)}^{3/2}}=0\end{array}$$

Solving the above for $p$ results in

$$\begin{array}{rl}{p}_1& =-1\end{array}$$

Substituting these in (1A) and keeping singular solution that verifies the ode gives

$$\begin{array}{r}p=-y\end{array}$$

The general solution is found when $\frac{\mathrm{d}p}{\mathrm{d}y}\ne 0$. From eq. (2A). This results in

$$\begin{array}{}\text{(3)}& p^{\prime }\left(y\right)=-\frac{2(p\left(y\right)+\frac{1}{{(-p\left(y\right))}^{3/2}}){(-p\left(y\right))}^{5/2}}{3y}\end{array}$$

This ODE is now solved for $p\left(y\right)$. No inversion is needed.

The ode

$$\begin{array}{}\text{(2)}& p^{\prime }\left(y\right)=-\frac{2({(-p\left(y\right))}^{5/2}-1)p\left(y\right)}{3y}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}{p}^{\prime }\left(y\right)& =-\frac{2({(-p\left(y\right))}^{5/2}-1)p\left(y\right)}{3y}\\ & =f(y)g(p)\end{array}$$

Where

$$\begin{array}{rl}f(y)& =-\frac{2}{3y}\\ g(p)& =({(-p)}^{5/2}-1)p\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(p)}dp& =\int f(y)dy\\ \int \frac{1}{({(-p)}^{5/2}-1)p}dp& =\int -\frac{2}{3y}dy\end{array}$$

$$\frac{2\ln (p{\left(y\right)}^2+{(-p\left(y\right))}^{3/2}-p\left(y\right)+\sqrt{-p\left(y\right)}+1)}{5}+\frac{2\ln (\sqrt{-p\left(y\right)}-1)}{5}-\ln (-p\left(y\right))=\ln \left(\frac{1}{y^{2/3}}\right)+c_2$$

Substituing the above solution for $p$ in (2A) gives

$$\begin{array}{rl}p& =-\frac{y}{{\left({({\mathrm{RootOf}((y^5-{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{75}+(15y^5-15{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{70}+(105y^5-105{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{65}+(455y^5-455{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{60}+(1365y^5-1365{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{55}+(3000y^5-3003{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{50}+(4975y^5-5005{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{45}+(6300y^5-6435{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{40}+(6075y^5-6435{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{35}+(4375y^5-5005{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{30}+(2250y^5-3003{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{25}+(750y^5-1365{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{20}+(125y^5-455{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{15}-105{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^{10}-15{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^5-{\mathrm{e}}^{\frac{15c_2}{2}})}^5+1)}^2\right)}^{3/2}}\end{array}$$

For solution (1) found earlier, since $p=y^{\prime }$ then we now have a new first order ode to solve which is

$$\begin{array}{r}{y}^{\prime }=0\end{array}$$

Since the ode has the form $y^{\prime }=f(x)$, then we only need to integrate $f(x)$.

$$\begin{array}{rl}\int dy& =\int 0dx+c_3\\ y& =c_3\end{array}$$

For solution (2) found earlier, since $p=y^{\prime }$ then we now have a new first order ode to solve which is

$$\begin{array}{r}{y}^{\prime }=\frac{y}{{\left({({\mathrm{RootOf}(-1+(y^5{\mathrm{e}}^{\frac{15c_1}{2}}+1){\mathit{\text{\_Z}}}^{75}+(-15y^5{\mathrm{e}}^{\frac{15c_1}{2}}-15){\mathit{\text{\_Z}}}^{70}+(105y^5{\mathrm{e}}^{\frac{15c_1}{2}}+105){\mathit{\text{\_Z}}}^{65}+(-455y^5{\mathrm{e}}^{\frac{15c_1}{2}}-455){\mathit{\text{\_Z}}}^{60}+(1365y^5{\mathrm{e}}^{\frac{15c_1}{2}}+1365){\mathit{\text{\_Z}}}^{55}+(-3000y^5{\mathrm{e}}^{\frac{15c_1}{2}}-3003){\mathit{\text{\_Z}}}^{50}+(4975y^5{\mathrm{e}}^{\frac{15c_1}{2}}+5005){\mathit{\text{\_Z}}}^{45}+(-6300y^5{\mathrm{e}}^{\frac{15c_1}{2}}-6435){\mathit{\text{\_Z}}}^{40}+(6075y^5{\mathrm{e}}^{\frac{15c_1}{2}}+6435){\mathit{\text{\_Z}}}^{35}+(-4375y^5{\mathrm{e}}^{\frac{15c_1}{2}}-5005){\mathit{\text{\_Z}}}^{30}+(2250y^5{\mathrm{e}}^{\frac{15c_1}{2}}+3003){\mathit{\text{\_Z}}}^{25}+(-750y^5{\mathrm{e}}^{\frac{15c_1}{2}}-1365){\mathit{\text{\_Z}}}^{20}+(125y^5{\mathrm{e}}^{\frac{15c_1}{2}}+455){\mathit{\text{\_Z}}}^{15}-105{\mathit{\text{\_Z}}}^{10}+15{\mathit{\text{\_Z}}}^5)}^5-1)}^2\right)}^{3/2}}\end{array}$$

Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as

$${\int }_{}^y\frac{{\left({({\mathrm{RootOf}(-1+({\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+1){\mathit{\text{\_Z}}}^{75}+(-15{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-15){\mathit{\text{\_Z}}}^{70}+(105{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+105){\mathit{\text{\_Z}}}^{65}+(-455{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-455){\mathit{\text{\_Z}}}^{60}+(1365{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+1365){\mathit{\text{\_Z}}}^{55}+(-3000{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-3003){\mathit{\text{\_Z}}}^{50}+(4975{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+5005){\mathit{\text{\_Z}}}^{45}+(-6300{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-6435){\mathit{\text{\_Z}}}^{40}+(6075{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+6435){\mathit{\text{\_Z}}}^{35}+(-4375{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-5005){\mathit{\text{\_Z}}}^{30}+(2250{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+3003){\mathit{\text{\_Z}}}^{25}+(-750{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-1365){\mathit{\text{\_Z}}}^{20}+(125{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+455){\mathit{\text{\_Z}}}^{15}-105{\mathit{\text{\_Z}}}^{10}+15{\mathit{\text{\_Z}}}^5)}^5-1)}^2\right)}^{3/2}}{\tau }d\tau =x+c_4$$

For solution (3) found earlier, since $p=y^{\prime }$ then we now have a new first order ode to solve which is

$$\begin{array}{r}{y}^{\prime }=-y\end{array}$$

Integrating gives

$$\begin{array}{rl}\int -\frac{1}{y}dy& =dx\\ -\ln \left(y\right)& =x+c_5\end{array}$$

Singular solutions are found by solving

$$\begin{array}{rl}-y& =0\end{array}$$

for $y$. This is because we had to divide by this in the above step. This gives the following singular solution(s), which also have to satisfy the given ODE.

$$\begin{array}{r}y=0\end{array}$$

For solution (4) found earlier, since $p=y^{\prime }$ then we now have a new first order ode to solve which is

$$\begin{array}{r}{y}^{\prime }=-\frac{y}{{\left({({\mathrm{RootOf}(-(y^5-{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{75}-(15y^5-15{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{70}-(105y^5-105{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{65}-(455y^5-455{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{60}-(1365y^5-1365{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{55}-(3000y^5-3003{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{50}-(4975y^5-5005{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{45}-(6300y^5-6435{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{40}-(6075y^5-6435{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{35}-(4375y^5-5005{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{30}-(2250y^5-3003{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{25}-(750y^5-1365{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{20}-(125y^5-455{\mathrm{e}}^{\frac{15c_2}{2}}){\mathit{\text{\_Z}}}^{15}+105{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^{10}+15{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^5+{\mathrm{e}}^{\frac{15c_2}{2}})}^5+1)}^2\right)}^{3/2}}\end{array}$$

Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as

$${\int }_{}^y-\frac{{\left({({\mathrm{RootOf}((-{\mathrm{e}}^{\frac{15c_2}{2}}+{\tau }^5){\mathit{\text{\_Z}}}^{75}+(-15{\mathrm{e}}^{\frac{15c_2}{2}}+15{\tau }^5){\mathit{\text{\_Z}}}^{70}+(-105{\mathrm{e}}^{\frac{15c_2}{2}}+105{\tau }^5){\mathit{\text{\_Z}}}^{65}+(-455{\mathrm{e}}^{\frac{15c_2}{2}}+455{\tau }^5){\mathit{\text{\_Z}}}^{60}+(-1365{\mathrm{e}}^{\frac{15c_2}{2}}+1365{\tau }^5){\mathit{\text{\_Z}}}^{55}+(-3003{\mathrm{e}}^{\frac{15c_2}{2}}+3000{\tau }^5){\mathit{\text{\_Z}}}^{50}+(-5005{\mathrm{e}}^{\frac{15c_2}{2}}+4975{\tau }^5){\mathit{\text{\_Z}}}^{45}+(-6435{\mathrm{e}}^{\frac{15c_2}{2}}+6300{\tau }^5){\mathit{\text{\_Z}}}^{40}+(-6435{\mathrm{e}}^{\frac{15c_2}{2}}+6075{\tau }^5){\mathit{\text{\_Z}}}^{35}+(-5005{\mathrm{e}}^{\frac{15c_2}{2}}+4375{\tau }^5){\mathit{\text{\_Z}}}^{30}+(-3003{\mathrm{e}}^{\frac{15c_2}{2}}+2250{\tau }^5){\mathit{\text{\_Z}}}^{25}+(-1365{\mathrm{e}}^{\frac{15c_2}{2}}+750{\tau }^5){\mathit{\text{\_Z}}}^{20}+(-455{\mathrm{e}}^{\frac{15c_2}{2}}+125{\tau }^5){\mathit{\text{\_Z}}}^{15}-105{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^{10}-15{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^5-{\mathrm{e}}^{\frac{15c_2}{2}})}^5+1)}^2\right)}^{3/2}}{\tau }d\tau =x+c_6$$

Will add steps showing solving for IC soon.

Solving for $y$ from the above solution(s) gives (after possible removing of solutions that do not verify)

$$\begin{array}{rl}{\int }_{}^y\frac{{\left({({\mathrm{RootOf}(-1+({\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+1){\mathit{\text{\_Z}}}^{75}+(-15{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-15){\mathit{\text{\_Z}}}^{70}+(105{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+105){\mathit{\text{\_Z}}}^{65}+(-455{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-455){\mathit{\text{\_Z}}}^{60}+(1365{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+1365){\mathit{\text{\_Z}}}^{55}+(-3000{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-3003){\mathit{\text{\_Z}}}^{50}+(4975{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+5005){\mathit{\text{\_Z}}}^{45}+(-6300{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-6435){\mathit{\text{\_Z}}}^{40}+(6075{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+6435){\mathit{\text{\_Z}}}^{35}+(-4375{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-5005){\mathit{\text{\_Z}}}^{30}+(2250{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+3003){\mathit{\text{\_Z}}}^{25}+(-750{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-1365){\mathit{\text{\_Z}}}^{20}+(125{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+455){\mathit{\text{\_Z}}}^{15}-105{\mathit{\text{\_Z}}}^{10}+15{\mathit{\text{\_Z}}}^5)}^5-1)}^2\right)}^{3/2}}{\tau }d\tau & =x+c_4\\ {\int }_{}^y-\frac{{\left({({\mathrm{RootOf}((-{\mathrm{e}}^{\frac{15c_2}{2}}+{\tau }^5){\mathit{\text{\_Z}}}^{75}+(-15{\mathrm{e}}^{\frac{15c_2}{2}}+15{\tau }^5){\mathit{\text{\_Z}}}^{70}+(-105{\mathrm{e}}^{\frac{15c_2}{2}}+105{\tau }^5){\mathit{\text{\_Z}}}^{65}+(-455{\mathrm{e}}^{\frac{15c_2}{2}}+455{\tau }^5){\mathit{\text{\_Z}}}^{60}+(-1365{\mathrm{e}}^{\frac{15c_2}{2}}+1365{\tau }^5){\mathit{\text{\_Z}}}^{55}+(-3003{\mathrm{e}}^{\frac{15c_2}{2}}+3000{\tau }^5){\mathit{\text{\_Z}}}^{50}+(-5005{\mathrm{e}}^{\frac{15c_2}{2}}+4975{\tau }^5){\mathit{\text{\_Z}}}^{45}+(-6435{\mathrm{e}}^{\frac{15c_2}{2}}+6300{\tau }^5){\mathit{\text{\_Z}}}^{40}+(-6435{\mathrm{e}}^{\frac{15c_2}{2}}+6075{\tau }^5){\mathit{\text{\_Z}}}^{35}+(-5005{\mathrm{e}}^{\frac{15c_2}{2}}+4375{\tau }^5){\mathit{\text{\_Z}}}^{30}+(-3003{\mathrm{e}}^{\frac{15c_2}{2}}+2250{\tau }^5){\mathit{\text{\_Z}}}^{25}+(-1365{\mathrm{e}}^{\frac{15c_2}{2}}+750{\tau }^5){\mathit{\text{\_Z}}}^{20}+(-455{\mathrm{e}}^{\frac{15c_2}{2}}+125{\tau }^5){\mathit{\text{\_Z}}}^{15}-105{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^{10}-15{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^5-{\mathrm{e}}^{\frac{15c_2}{2}})}^5+1)}^2\right)}^{3/2}}{\tau }d\tau & =x+c_6\\ y& =0\\ y& =c_3\\ y& ={\mathrm{e}}^{-x-c_5}\end{array}$$

The solution

$${\int }_{}^y\frac{{\left({({\mathrm{RootOf}(-1+({\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+1){\mathit{\text{\_Z}}}^{75}+(-15{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-15){\mathit{\text{\_Z}}}^{70}+(105{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+105){\mathit{\text{\_Z}}}^{65}+(-455{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-455){\mathit{\text{\_Z}}}^{60}+(1365{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+1365){\mathit{\text{\_Z}}}^{55}+(-3000{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-3003){\mathit{\text{\_Z}}}^{50}+(4975{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+5005){\mathit{\text{\_Z}}}^{45}+(-6300{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-6435){\mathit{\text{\_Z}}}^{40}+(6075{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+6435){\mathit{\text{\_Z}}}^{35}+(-4375{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-5005){\mathit{\text{\_Z}}}^{30}+(2250{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+3003){\mathit{\text{\_Z}}}^{25}+(-750{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}-1365){\mathit{\text{\_Z}}}^{20}+(125{\tau }^5{\mathrm{e}}^{\frac{15c_1}{2}}+455){\mathit{\text{\_Z}}}^{15}-105{\mathit{\text{\_Z}}}^{10}+15{\mathit{\text{\_Z}}}^5)}^5-1)}^2\right)}^{3/2}}{\tau }d\tau =x+c_4$$

was found not to satisfy the ode or the IC. Hence it is removed. The solution

$${\int }_{}^y-\frac{{\left({({\mathrm{RootOf}((-{\mathrm{e}}^{\frac{15c_2}{2}}+{\tau }^5){\mathit{\text{\_Z}}}^{75}+(-15{\mathrm{e}}^{\frac{15c_2}{2}}+15{\tau }^5){\mathit{\text{\_Z}}}^{70}+(-105{\mathrm{e}}^{\frac{15c_2}{2}}+105{\tau }^5){\mathit{\text{\_Z}}}^{65}+(-455{\mathrm{e}}^{\frac{15c_2}{2}}+455{\tau }^5){\mathit{\text{\_Z}}}^{60}+(-1365{\mathrm{e}}^{\frac{15c_2}{2}}+1365{\tau }^5){\mathit{\text{\_Z}}}^{55}+(-3003{\mathrm{e}}^{\frac{15c_2}{2}}+3000{\tau }^5){\mathit{\text{\_Z}}}^{50}+(-5005{\mathrm{e}}^{\frac{15c_2}{2}}+4975{\tau }^5){\mathit{\text{\_Z}}}^{45}+(-6435{\mathrm{e}}^{\frac{15c_2}{2}}+6300{\tau }^5){\mathit{\text{\_Z}}}^{40}+(-6435{\mathrm{e}}^{\frac{15c_2}{2}}+6075{\tau }^5){\mathit{\text{\_Z}}}^{35}+(-5005{\mathrm{e}}^{\frac{15c_2}{2}}+4375{\tau }^5){\mathit{\text{\_Z}}}^{30}+(-3003{\mathrm{e}}^{\frac{15c_2}{2}}+2250{\tau }^5){\mathit{\text{\_Z}}}^{25}+(-1365{\mathrm{e}}^{\frac{15c_2}{2}}+750{\tau }^5){\mathit{\text{\_Z}}}^{20}+(-455{\mathrm{e}}^{\frac{15c_2}{2}}+125{\tau }^5){\mathit{\text{\_Z}}}^{15}-105{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^{10}-15{\mathrm{e}}^{\frac{15c_2}{2}}{\mathit{\text{\_Z}}}^5-{\mathrm{e}}^{\frac{15c_2}{2}})}^5+1)}^2\right)}^{3/2}}{\tau }d\tau =x+c_6$$

was found not to satisfy the ode or the IC. Hence it is removed.

Summary of solutions found

$$\begin{array}{rl}y& =0\\ y& =c_3\\ y& ={\mathrm{e}}^{-x-c_5}\end{array}$$

✓ Maple. Time used: 0.184 (sec). Leaf size: 126

ode:=y(x)*diff(diff(y(x),x),x)^3+y(x)^3*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

$$\begin{array}{rl}y& =0\\ y& =c_1\\ y& ={\mathrm{e}}^{\int \mathrm{RootOf}(x-{\int }_{}^{\mathit{\text{\_Z}}}-\frac{1}{{\mathit{\text{\_f}}}^2-{(-\mathit{\text{\_f}})}^{1/3}}d\mathit{\text{\_f}}+c_1)dx+c_2}\\ y& ={\mathrm{e}}^{\int \mathrm{RootOf}(x+2\left({\int }_{}^{\mathit{\text{\_Z}}}\frac{1}{i{(-\mathit{\text{\_f}})}^{1/3}\sqrt{3}+2{\mathit{\text{\_f}}}^2+{(-\mathit{\text{\_f}})}^{1/3}}d\mathit{\text{\_f}}\right)+c_1)dx+c_2}\\ y& ={\mathrm{e}}^{\int \mathrm{RootOf}(x-2\left({\int }_{}^{\mathit{\text{\_Z}}}\frac{1}{i{(-\mathit{\text{\_f}})}^{1/3}\sqrt{3}-2{\mathit{\text{\_f}}}^2-{(-\mathit{\text{\_f}})}^{1/3}}d\mathit{\text{\_f}}\right)+c_1)dx+c_2}\end{array}$$

Maple trace

`Methods for second order ODEs: 
   *** Sublevel 2 *** 
   Methods for second order ODEs: 
   Successful isolation of d^2y/dx^2: 3 solutions were found. Trying to solve each resulting ODE. 
      *** Sublevel 3 *** 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying 2nd order Liouville 
      trying 2nd order WeierstrassP 
      trying 2nd order JacobiSN 
      differential order: 2; trying a linearization to 3rd order 
      trying 2nd order ODE linearizable_by_differentiation 
      trying 2nd order, 2 integrating factors of the form mu(x,y) 
      trying differential order: 2; missing variables 
      `, `-> Computing symmetries using: way = 3 
      Try integration with the canonical coordinates of the symmetry [0, y] 
      -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -_b(_a)^2+(-_b(_a))^(1/3), _b(_a), explicit, HINT = [[1, 0]]`         *** 
         symmetry methods on request 
      `, `1st order, trying reduction of order with given symmetries:`[1, 0]
 

✓ Mathematica. Time used: 2.742 (sec). Leaf size: 800

ode=y[x]*D[y[x],{x,2}]^3+y[x]^3*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$\begin{array}{r}\text{Solution too large to show}\end{array}$$

✓ Sympy. Time used: 0.253 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)**3*Derivative(y(x), x) + y(x)*Derivative(y(x), (x, 2))**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

$$y\left(x\right)=0$$