Internal problem ID [4303]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 36
Problem number: 1088.
ODE order: 1.
ODE degree: 4.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3}=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 262
dsolve(diff(y(x),x)^4+f(x)*(y(x)-a)^3*(y(x)-b)^3 = 0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {3}{4}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{3}\right )^{\frac {1}{4}}d \textit {\_a}}{\left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {3}{4}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\frac {i \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{3}\right )^{\frac {1}{4}}d \textit {\_a} \right )}{\left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {3}{4}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {3}{4}}}d \textit {\_a} -\frac {i \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{3}\right )^{\frac {1}{4}}d \textit {\_a} \right )}{\left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {3}{4}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{3}\right )^{\frac {1}{4}}d \textit {\_a}}{\left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {3}{4}}}+c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.126 (sec). Leaf size: 385
DSolve[(y'[x])^4 +f[x] (y[x]-a)^3 (y[x]-b)^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\&\right ]\left [\int _1^x-\sqrt [4]{-1} \sqrt [4]{f(K[1])}dK[1]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\&\right ]\left [\int _1^x\sqrt [4]{-1} \sqrt [4]{f(K[2])}dK[2]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\&\right ]\left [\int _1^x-(-1)^{3/4} \sqrt [4]{f(K[3])}dK[3]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\&\right ]\left [\int _1^x(-1)^{3/4} \sqrt [4]{f(K[4])}dK[4]+c_1\right ] \\ y(x)\to a \\ y(x)\to b \\ \end{align*}