7.2.21 problem 23

Internal problem ID [39]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.3. Problems at page 27
Problem number : 23
Date solved : Friday, February 07, 2025 at 08:12:39 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=x^{2}+y^{2}-1 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \end{align*}

✗ Solution by Maple

dsolve([diff(y(x),x)=x^2+y(x)^2-1,y(0) = 0],y(x), singsol=all)
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 0.215 (sec). Leaf size: 178

DSolve[{D[y[x],x]==x^2+y[x]^2-1,{y[0]==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {2^{1 i/2} x \operatorname {Gamma}\left (\frac {1}{4}+\frac {i}{4}\right ) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}-\frac {i}{2},(-1+i) x\right )+i x \operatorname {Gamma}\left (\frac {1}{4}-\frac {i}{4}\right ) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}+\frac {i}{2},(1+i) x\right )+(1+i) \left (2^{1 i/2} \operatorname {Gamma}\left (\frac {1}{4}+\frac {i}{4}\right ) \operatorname {ParabolicCylinderD}\left (\frac {1}{2}-\frac {i}{2},(-1+i) x\right )-\operatorname {Gamma}\left (\frac {1}{4}-\frac {i}{4}\right ) \operatorname {ParabolicCylinderD}\left (\frac {1}{2}+\frac {i}{2},(1+i) x\right )\right )}{i 2^{1 i/2} \operatorname {Gamma}\left (\frac {1}{4}+\frac {i}{4}\right ) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}-\frac {i}{2},(-1+i) x\right )+\operatorname {Gamma}\left (\frac {1}{4}-\frac {i}{4}\right ) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}+\frac {i}{2},(1+i) x\right )} \]