Internal problem ID [8338]
Internal file name [OUTPUT/7271_Sunday_June_05_2022_05_41_41_PM_7256698/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 1.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }=\frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}}} \]
Integrating both sides gives \begin {align*} y = \int \frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}}d x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \int \frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}}d x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = \int \frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}}d x +c_{1} \] Verified OK.
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 30
dsolve(diff(y(x),x) - (a4*x^4+a3*x^3+a2*x^2+a1*x+a0)^(-1/2)=0,y(x), singsol=all)
\[ y \left (x \right ) = \int \frac {1}{\sqrt {\operatorname {a4} \,x^{4}+\operatorname {a3} \,x^{3}+\operatorname {a2} \,x^{2}+\operatorname {a1} x +\operatorname {a0}}}d x +c_{1} \]
✓ Solution by Mathematica
Time used: 10.268 (sec). Leaf size: 1117
DSolve[y'[x] - (a4*x^4+a3*x^3+a2*x^2+a1*x+a0)^(-1/2)==0,y[x],x,IncludeSingularSolutions -> True]
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