MathAction Solving Differential Equations

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Test 1: solve a simple nonlinear homogeneous differential equation

fricas

y := operator y

$\label{eq1}y$

(1)

Type: BasicOperator?

fricas

deq1 := D(y(x),x) = 9 - y(x)^2

$\label{eq2}{{y^{\prime}}\left({x}\right)}={-{{y \left({x}\right)}^{2}}+ 9}$

(2)

Type: Equation(Expression(Integer))

fricas

solve(deq1,y,x)

$\label{eq3}{-{\log \left({{y \left({x}\right)}+ 3}\right)}+{\log \left({{y \left({x}\right)}- 3}\right)}+{6 \ x}}\over 6$

(3)

Type: Union(Expression(Integer),...)

Test 2: solve a class of simple nonlinear homogeneous differential equations

fricas

deq2a := D(y(x),x) = c - p*y(x)^2

$\label{eq4}{{y^{\prime}}\left({x}\right)}={-{p \ {{y \left({x}\right)}^{2}}}+ c}$

(4)

Type: Equation(Expression(Integer))

fricas

xpr2b := solve(deq2a,y,x)

$\label{eq5}{{\log \left({{{{\left({p \ {{y \left({x}\right)}^{2}}}+ c \right)}\ {\sqrt{c \ p}}}-{2 \ c \ p \ {y \left({x}\right)}}}\over{{p \ {{y \left({x}\right)}^{2}}}- c}}\right)}+{2 \ x \ {\sqrt{c \ p}}}}\over{2 \ {\sqrt{c \ p}}}$

(5)

Type: Union(Expression(Integer),...)

fricas

simplify(x-xpr2b)

$\label{eq6}-{{\log \left({{{{\left({p \ {{y \left({x}\right)}^{2}}}+ c \right)}\ {\sqrt{c \ p}}}-{2 \ c \ p \ {y \left({x}\right)}}}\over{{p \ {{y \left({x}\right)}^{2}}}- c}}\right)}\over{2 \ {\sqrt{c \ p}}}}$

(6)

Type: Expression(Integer)

Test 3: find general solutions for nonlinear homogeneous differential equations

fricas

f := operator f

$\label{eq7}f$

(7)

Type: BasicOperator?

fricas

deq3 := D(y(x),x) = f(y(x))

$\label{eq8}{{y^{\prime}}\left({x}\right)}={f \left({y \left({x}\right)}\right)}$

(8)

Type: Equation(Expression(Integer))

fricas

solve(deq3,y,x)

$\label{eq9}{\int^{ \displaystyle {y \left({x}\right)}}{{1 \over{f \left({\%D}\right)}}\ {d \%D}}}- x$

(9)

Type: Union(Expression(Integer),...)

Test 4: integration

fricas

integrate(1/(1-x^2),x)

$\label{eq10}{{\log \left({x + 1}\right)}-{\log \left({x - 1}\right)}}\over 2$

(10)

Type: Union(Expression(Integer),...)

Test 5: check result

fricas

xpr5 := (log(x+1)-log(x-1))/2

$\label{eq11}{{\log \left({x + 1}\right)}-{\log \left({x - 1}\right)}}\over 2$

(11)

Type: Expression(Integer)

fricas

D(xpr5,x)

$\label{eq12}-{1 \over{{{x}^{2}}- 1}}$

(12)

Type: Expression(Integer)

Test 6: check simplification

fricas

xpr6 := log(1+2/(x-1))/2

$\label{eq13}{\log \left({{x + 1}\over{x - 1}}\right)}\over 2$

(13)

Type: Expression(Integer)

fricas

D(xpr6,x)

$\label{eq14}-{1 \over{{{x}^{2}}- 1}}$

(14)

Type: Expression(Integer)

Test 7: Express y(x) as function of x (replacing y(x) with z)

fricas

eq7a := x = (log(z+1)-log(z-1))/2

$\label{eq15}x ={{{\log \left({z + 1}\right)}-{\log \left({z - 1}\right)}}\over 2}$

(15)

Type: Equation(Expression(Integer))

fricas

solve(eq7a,z)

$\label{eq16}\left[{z ={{-{{e}^{-{2 \ x}}}- 1}\over{{{e}^{-{2 \ x}}}- 1}}}\right]$

(16)

Type: List(Equation(Expression(Integer)))

fricas

xpr7b := (1+exp(-2*z))/(1-exp(-2*z))

$\label{eq17}{-{{e}^{-{2 \ z}}}- 1}\over{{{e}^{-{2 \ z}}}- 1}$

(17)

Type: Expression(Integer)

fricas

simplify(xpr7b)

$\label{eq18}{-{{e}^{-{2 \ z}}}- 1}\over{{{e}^{-{2 \ z}}}- 1}$

(18)

Type: Expression(Integer)

Test 8: check simplified result

fricas

xpr8 := (1+exp(-2*x))/(1-exp(-2*x)) - (1+2/(exp(2*x)-1))

$\label{eq19}{-{2 \ {{e}^{-{2 \ x}}}\ {{e}^{2 \ x}}}+ 2}\over{{{\left({{e}^{-{2 \ x}}}- 1 \right)}\ {{e}^{2 \ x}}}-{{e}^{-{2 \ x}}}+ 1}$

(19)

Type: Expression(Integer)

fricas

simplify(xpr8)

$\label{eq20}0$

(20)

Type: Expression(Integer)

Test 9: check result by substitution in the DEQ

fricas

xpr9a := (1+2/(exp(2*x)-1))

$\label{eq21}{{{e}^{2 \ x}}+ 1}\over{{{e}^{2 \ x}}- 1}$

(21)

Type: Expression(Integer)

fricas

xpr9b := D(xpr9a,x)

$\label{eq22}-{{4 \ {{e}^{2 \ x}}}\over{{{{e}^{2 \ x}}^{2}}-{2 \ {{e}^{2 \ x}}}+ 1}}$

(22)

Type: Expression(Integer)

fricas

xpr9c := 1 - (xpr9a)^2

$\label{eq23}-{{4 \ {{e}^{2 \ x}}}\over{{{{e}^{2 \ x}}^{2}}-{2 \ {{e}^{2 \ x}}}+ 1}}$

(23)

Type: Expression(Integer)

Test 10: finding the explicit solution for the deq in test 2

fricas

wcp := sqrt(c*p)

$\label{eq24}\sqrt{c \ p}$

(24)

Type: Expression(Integer)

fricas

xpr10a := log(((p*z^2+c)*wcp-2*c*p*z)/(p*z^2-c))/(2*wcp)

$\label{eq25}{\log \left({{{{\left({p \ {{z}^{2}}}+ c \right)}\ {\sqrt{c \ p}}}-{2 \ c \ p \ z}}\over{{p \ {{z}^{2}}}- c}}\right)}\over{2 \ {\sqrt{c \ p}}}$

(25)

Type: Expression(Integer)

fricas

solve(x = xpr10a,z)

$\label{eq26}\begin{array}{@{}l} \displaystyle \left[{z ={{-{{\sqrt{c \ p}}\ {{{e}^{2 \ x \ {\sqrt{c \ p}}}}^{2}}}-{2 \ c \ p \ {{e}^{2 \ x \ {\sqrt{c \ p}}}}}-{c \ p \ {\sqrt{c \ p}}}}\over{{p \ {{{e}^{2 \ x \ {\sqrt{c \ p}}}}^{2}}}-{c \ {{p}^{2}}}}}}, \: \right. \ \ \displaystyle \left.{z ={{\sqrt{c \ p}}\over p}}\right]$

(26)

Type: List(Equation(Expression(Integer)))

fricas

eq10c := (p*z^2+c)*wcp-2*c*p*z

$\label{eq27}{{\left({p \ {{z}^{2}}}+ c \right)}\ {\sqrt{c \ p}}}-{2 \ c \ p \ z}$

(27)

Type: Expression(Integer)

fricas

solve(eq10c=0,z)

$\label{eq28}\left[{z ={{\sqrt{c \ p}}\over p}}\right]$

(28)

Type: List(Equation(Expression(Integer)))

fricas

xpr10d := p*wcp*(z-wcp/p)^2

$\label{eq29}{{\left({p \ {{z}^{2}}}+ c \right)}\ {\sqrt{c \ p}}}-{2 \ c \ p \ z}$

(29)

Type: Expression(Integer)

fricas

simplify(xpr10d/eq10c)

$\label{eq30}1$

(30)

Type: Expression(Integer)

fricas

xpr10e := (wcp/p)*(2/(1-exp(-2*wcp*x)/wcp)-1)

$\label{eq31}{-{{\sqrt{c \ p}}\ {{e}^{-{2 \ x \ {\sqrt{c \ p}}}}}}-{c \ p}}\over{{p \ {{e}^{-{2 \ x \ {\sqrt{c \ p}}}}}}-{p \ {\sqrt{c \ p}}}}$

(31)

Type: Expression(Integer)

fricas

xpr10f := D(xpr10e,x)

$\label{eq32}-{{4 \ c \ {\sqrt{c \ p}}\ {{e}^{-{2 \ x \ {\sqrt{c \ p}}}}}}\over{{{{e}^{-{2 \ x \ {\sqrt{c \ p}}}}}^{2}}-{2 \ {\sqrt{c \ p}}\ {{e}^{-{2 \ x \ {\sqrt{c \ p}}}}}}+{c \ p}}}$

(32)

Type: Expression(Integer)

fricas

xpr10g := c - p*(xpr10e)^2

$\label{eq33}-{{4 \ c \ {\sqrt{c \ p}}\ {{e}^{-{2 \ x \ {\sqrt{c \ p}}}}}}\over{{{{e}^{-{2 \ x \ {\sqrt{c \ p}}}}}^{2}}-{2 \ {\sqrt{c \ p}}\ {{e}^{-{2 \ x \ {\sqrt{c \ p}}}}}}+{c \ p}}}$

(33)

Type: Expression(Integer)

fricas

simplify(xpr10f / xpr10g)

$\label{eq34}1$

(34)

Type: Expression(Integer)

Test 11: non-homogeneous generalization: c as linear function of x

fricas

deq11a := D(y(x),x) = a*x + b - p*y(x)^2

$\label{eq35}{{y^{\prime}}\left({x}\right)}={-{p \ {{y \left({x}\right)}^{2}}}+{a \ x}+ b}$

(35)

Type: Equation(Expression(Integer))

fricas

solve(deq11a,y,x)

$\label{eq36}\verb#"failed"#$

(36)

Type: Union("failed",...)

Test 12: non-homogeneous generalization: c as arbitrary function of x

fricas

c := operator c

$\label{eq37}c$

(37)

Type: BasicOperator?

fricas

deq12a := D(y(x),x) = c(x) - p*y(x)^2

$\label{eq38}{{y^{\prime}}\left({x}\right)}={-{p \ {{y \left({x}\right)}^{2}}}+{c \left({x}\right)}}$

(38)

Type: Equation(Expression(Integer))

fricas

solve(deq12a,y,x)

$\label{eq39}\verb#"failed"#$

(39)

Type: Union("failed",...)

Test 13: guessing solution for deq12a

fricas

wpcx := sqrt(c(x))*sqrt(p)

$\label{eq40}{\sqrt{p}}\ {\sqrt{c \left({x}\right)}}$

(40)

Type: Expression(Integer)

fricas

xpr13a := log((p*wpcx*(z-wpcx/p)^2)/(p*z^2-c(x)))/(2*wpcx)

$\label{eq41}{\log \left({{{{\left(-{c \left({x}\right)}-{p \ {{z}^{2}}}\right)}\ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}+{2 \ p \ z \ {c \left({x}\right)}}}\over{{c \left({x}\right)}-{p \ {{z}^{2}}}}}\right)}\over{2 \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}$

(41)

Type: Expression(Integer)

fricas

xpr13b := (wpcx/p)*(2/(1-exp(-2*wpcx*x)/wpcx)-1)

$\label{eq42}{-{{\sqrt{p}}\ {\sqrt{c \left({x}\right)}}\ {{e}^{-{2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}}-{p \ {c \left({x}\right)}}}\over{{p \ {{e}^{-{2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}}-{p \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}$

(42)

Type: Expression(Integer)

fricas

xpr13c := D(xpr13b,x)

$\label{eq43}{\left( \begin{array}{@{}l} \displaystyle -{{\sqrt{p}}\ {{c^{\prime}}\left({x}\right)}\ {{{e}^{-{2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}^{2}}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle - \ \ \displaystyle {2 \ p \ {\sqrt{c \left({x}\right)}}}- \ \ \displaystyle {4 \ p \ x \ {c \left({x}\right)}\ {\sqrt{p}}}$

(43)

Type: Expression(Integer)

fricas

xpr13d := c(x) - p*(xpr13b)^2

$\label{eq44}-{{4 \ {c \left({x}\right)}\ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}\ {{e}^{-{2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}}\over{{{{e}^{-{2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}^{2}}-{2 \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}\ {{e}^{-{2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}}+{p \ {c \left({x}\right)}}}}$

(44)

Type: Expression(Integer)

fricas

simplify(xpr13d / xpr13c)

$\label{eq45}{\left( \begin{array}{@{}l} \displaystyle {8 \ {{p}^{2}}\ {{c \left({x}\right)}^{3}}\ {\sqrt{p}}\ {{e}^{-{2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}}- \ \ \displaystyle {{16}\ {{p}^{2}}\ {{c \left({x}\right)}^{2}}\ {\sqrt{c \left({x}\right)}}\ {{e}^{-{4 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}}+ \ \ \displaystyle {8 \ p \ {{c \left({x}\right)}^{2}}\ {\sqrt{p}}\ {{e}^{-{6 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}}$

(45)

Type: Expression(Integer)

fricas

xpr13e := simplify(D((2/(1-exp(-2*wpcx*x)/wpcx)-1),x))

$\label{eq46}{\left({\left({ \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {2 \ p \ x \ {\sqrt{c \left({x}\right)}}}+ \ \ \displaystyle {\sqrt{p}}$

(46)

Type: Expression(Integer)

fricas

xpr13f := (wpcx-exp(-2*wpcx*x))^2

$\label{eq47}{{{e}^{-{2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}^{2}}-{2 \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}\ {{e}^{-{2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}}}}+{p \ {c \left({x}\right)}}$

(47)

Type: Expression(Integer)

fricas

xpr13g := simplify(((2*p*c(x)*x+wpcx)*D(c(x),x)+4*p*c(x)^2)/wpcx)

$\label{eq48}{{{\left({{\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}+{2 \ p \ x \ {c \left({x}\right)}}\right)}\ {{c^{\prime}}\left({x}\right)}}+{4 \ p \ {{c \left({x}\right)}^{2}}}}\over{{\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}$

(48)

Type: Expression(Integer)

fricas

xpr13h := simplify((1+2*wpcx*x)*D(c(x),x)+4*wpcx*c(x))

$\label{eq49}{{\left({2 \ x \ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}+ 1 \right)}\ {{c^{\prime}}\left({x}\right)}}+{4 \ {c \left({x}\right)}\ {\sqrt{p}}\ {\sqrt{c \left({x}\right)}}}$

(49)

Type: Expression(Integer)

fricas

simplify(xpr13h/xpr13g)

$\label{eq50}1$

(50)

Type: Expression(Integer)

fricas

xpr13i := simplify((1/(wpcx-z)-1/(2*wpcx))*D(c(x),x) + ((1+2*wpcx*x)*D(c(x),x)+4*wpcx*c(x))*z/(wpcx-z)^2)

$\label{eq51}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle {\left({{\left(-{4 \ p \ x \ z}- p \right)}\ {c \left({x}\right)}}+{3 \ {{z}^{2}}}\right)}\ {\sqrt{p}}\ \cdot \ \ \displaystyle {\sqrt{c \left({x}\right)}}$

(51)

Type: Expression(Integer)

fricas

w := operator w

$\label{eq52}w$

(52)

Type: BasicOperator?

fricas

z := operator z

$\label{eq53}z$

(53)

Type: BasicOperator?

fricas

simplify(D((2/(1-z(x)/w(x))-1)*c(x)/w(x),x))

$\label{eq54}{\left( \begin{array}{@{}l} \displaystyle {2 \ {c \left({x}\right)}\ {{w \left({x}\right)}^{2}}\ {{z^{\prime}}\left({x}\right)}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{c \left({x}\right)}\ {{z \left({x}\right)}^{2}}}-{2 \ {c \left({x}\right)}\ {w \left({x}\right)}\ {z \left({x}\right)}}- \ \ \displaystyle {{c \left({x}\right)}\ {{w \left({x}\right)}^{2}}}$

(54)

Type: Expression(Integer)

fricas

simplify((2*c(x)*w(x)^2*D(z(x),x)+(z(x)^2-2*w(x)*z(x)-w(x)^2)*c(x)*D(w(x),x)+(w(x)^3-w(x)*z(x)^2)*D(c(x),x))/(w(x)^2*(z(x)-w(x))^2))

$\label{eq55}{\left( \begin{array}{@{}l} \displaystyle {2 \ {c \left({x}\right)}\ {{w \left({x}\right)}^{2}}\ {{z^{\prime}}\left({x}\right)}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{c \left({x}\right)}\ {{z \left({x}\right)}^{2}}}-{2 \ {c \left({x}\right)}\ {w \left({x}\right)}\ {z \left({x}\right)}}- \ \ \displaystyle {{c \left({x}\right)}\ {{w \left({x}\right)}^{2}}}$

(55)

Type: Expression(Integer)

fricas

xpr13j := simplify((2*c(x)*w(x)^2*w(x)*z(x)*(x/c(x)-1/2)+(z(x)^2-2*w(x)*z(x)-w(x)^2)*c(x)*(-w(x)/(2*c(x)))+(w(x)^3-w(x)*z(x)^2)*D(c(x),x))/(w(x)^2*(z(x)-w(x))^2))

$\label{eq56}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{2 \ {{z \left({x}\right)}^{2}}}+{2 \ {{w \left({x}\right)}^{2}}}\right)}\ {{c^{\prime}}\left({x}\right)}}-{{z \left({x}\right)}^{2}}+ \ \ \displaystyle {{\left({{\left(-{2 \ {c \left({x}\right)}}+{4 \ x}\right)}\ {{w \left({x}\right)}^{2}}}+{2 \ {w \left({x}\right)}}\right)}\ {z \left({x}\right)}}+{{w \left({x}\right)}^{2}}$

(56)

Type: Expression(Integer)

fricas

xpr13k := simplify((1/(w(x)-z(x))-1/(2*w(x)))*D(c(x),x) + ((1+2*w(x)*x)*D(c(x),x) + 4*w(x)*c(x))*z(x)/(w(x)-z(x))^2)

$\label{eq57}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{{z \left({x}\right)}^{2}}+{{\left({ \begin{array}{@{}l} \displaystyle {4 \ x \ {{w \left({x}\right)}^{2}}}+ \ \ \displaystyle {2 \ {w \left({x}\right)}}$

(57)

Type: Expression(Integer)

fricas

simplify(xpr13k/xpr13j)

$\label{eq58}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{z \left({x}\right)}^{2}}+{{\left({ \begin{array}{@{}l} \displaystyle -{4 \ x \ {{w \left({x}\right)}^{2}}}- \ \ \displaystyle {2 \ {w \left({x}\right)}}$

(58)

Type: Expression(Integer)

fricas

simplify(xpr13k-xpr13j)

$\label{eq59}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{z \left({x}\right)}^{2}}+{{\left({4 \ x \ {{w \left({x}\right)}^{2}}}+{2 \ {w \left({x}\right)}}\right)}\ {z \left({x}\right)}}- \ \ \displaystyle {{w \left({x}\right)}^{2}}$

(59)

Type: Expression(Integer)