dsolve(diff(y(x), x, x) = y(x))
f := diff(y(x), x, x)+a^2*y(x);
Maple:
dsolve(f);
dsolve(diff(y(x), x, x, x, x) = y(x));
y(x) = _C1*exp(x)+_C2*exp(-x)+_C3*sin(x)+_C4*cos(x)
dsolve(diff(y(x), x, x, x, x, x, x, x) = diff(y(x), x, x))
y(x) = _C1+_C2*x+_C3*exp(x)-_C4*exp((-1/4-(1/4)*sqrt(5))*x)*sin((1/4)*sqrt(2)*sqrt(5-sqrt(5))*x)-_C5*exp((-1/4+(1/4)*sqrt(5))*x)*sin((1/4)*sqrt(2)*sqrt(5+sqrt(5))*x)+_C6*exp((-1/4-(1/4)*sqrt(5))*x)*cos((1/4)*sqrt(2)*sqrt(5-sqrt(5))*x)+_C7*exp((-1/4+(1/4)*sqrt(5))*x)*cos((1/4)*sqrt(2)*sqrt(5+sqrt(5))*x)
Maple 式:
dsolve(f);
y(x) = e/d+_C1*exp((1/6)*((36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(2/3)-12*c*a+4*b^2-2*b*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(1/3))*x/(a*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(1/3)))+_C2*exp(-(1/12*I)*(-I*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(2/3)+(12*I)*c*a-(4*I)*b^2-(4*I)*b*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(1/3)+sqrt(3)*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(2/3)+12*sqrt(3)*c*a-4*sqrt(3)*b^2)*x/(a*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(1/3)))+_C3*exp((1/12*I)*(I*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(2/3)-(12*I)*c*a+(4*I)*b^2+(4*I)*b*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(1/3)+sqrt(3)*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(2/3)+12*sqrt(3)*c*a-4*sqrt(3)*b^2)*x/(a*(36*c*b*a-108*d*a^2-8*b^3+12*sqrt(3)*sqrt(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)*a)^(1/3)))
- Maple 可以直接得出解答,无需用户作分离变数步骤。
![{\displaystyle {\frac {dy(x)}{dx}}={\frac {1+y(x)^{2}}{1+x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6ae6bcdad945c690b39e81c269a74880b0f737)
- f := diff(y(x), x) = (1+y(x)^2)/(1+x^2);
- dsolve(f);
![{\displaystyle y(x)=tan(arctan(x)+_{C}1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f54ca6c63bcbf8072a22a7065965ae0475d0d04)
- f := diff(y(x), x) = (1+y(x)^2)*(1+1/x)/(1+x^2)
- dsolve(f);
![{\displaystyle y(x)=tan(ln(x)-(1/2)*ln(1+x^{2})+arctan(x)+_{C}1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38db0433dc7a40fec921f6f958de6918216cbe0e)
- f := diff(x(t), t) = (x(t)^2+t^2)/(t*x(t))
,
![{\displaystyle x(t)=-{\sqrt {2*ln(t)+_{C}1}}*t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68af07de71fcd7746ddcf59c05f6c762ec61ea64)
![{\displaystyle x(t)=-(1/5)*t-(1/5)*{\sqrt {t^{2}-10*t+10*_{C}1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35fd7231541af6a9fda1b4c6178ee5ad9094582a)
![{\displaystyle x(t)=-(1/5)*t+(1/5)*{\sqrt {t^{2}-10*t+10*_{C}1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37d7d3106641a0b4aadb16b82b457688419b73d3)
![{\displaystyle {\frac {dx}{dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd8f4577337ea1e3aa08b83b86f156fc04aff2c)
f := diff(x(t), t) = 5*x(t)/t+t*x(t)^(1/2)
![{\displaystyle {\sqrt {x(t)}}+t^{2}-t^{(}5/2)*_{C}1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13fde51ec0bbde325d34ca018d8a9ac044dbba5f)
- f := diff(x(t), t) = sin(t)*x(t)^2+2*sin(t)/cos(t)^2
![{\displaystyle x(t)=-(1/14)*(-{\sqrt {(}}7)+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50cd153dc3569f958e8f420510410dd02c19d6c8)
![{\displaystyle 7*tan((1/2*(ln(cos(t))+_{C}1))*{\sqrt {(}}7)))*{\sqrt {(}}7)/cos(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2140327b439ba480e3301d156bba6fe51fef453b)
- f := diff(x(t), t) = t^5*x(t)^2+5*x(t)/t+6/t^2
![{\displaystyle x(t)={\frac {-{\sqrt {(}}6)*(BesselJ(6/5,(2/5)*{\sqrt {(}}6)*t^{(}5/2))+_{C}1*BesselY(6/5,(2/5)*{\sqrt {(}}6)*t^{(}5/2)))}{(t^{(}7/2)*(_{C}1*BesselY(11/5,(2/5)*{\sqrt {(}}6)*t^{(}5/2))+BesselJ(11/5,(2/5)*{\sqrt {(}}6)*t^{(}5/2))))}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d876f5c4d0e7674bbc220bb54f8d516b54f1331)
f := (1-x^2)*(diff(y(x), x, x))-2*x*(diff(y(x), x))+l*(l+1);
> dsolve(f);
y(x) = (1/2)*ln(x-1)*_C1+(1/2)*ln(x-1)*l^2+(1/2)*ln(x-1)*l-(1/2)*ln(x+1)*_C1+(1/2)*ln(x+1)*l^2+(1/2)*ln(x+1)*l+_C2
f := x^2*(diff(y(x), x, x))+x*(diff(y(x), x))+(x^2-v^2)*y(x)
dsolve(f);
y(x) = _C1*BesselJ(v, x)+_C2*BesselY(v, x)
f := (1-x^2)*(diff(y(x), x, x))-2*x*(diff(y(x), x))+l(l+1)*y(x);
dsolve(f);
y(x) = _C1*LegendreP((1/2)*sqrt(1+4*l(l+1))-1/2, x)+_C2*LegendreQ((1/2)*sqrt(1+4*l(l+1))-1/2, x)
f := diff(y(x), x, x)-x*(diff(y(x), x))+n*y(x);
dsolve(f);
y(x) = _C1*KummerM(1/2-(1/2)*n, 3/2, (1/2)*x^2)*x+_C2*KummerU(1/2-(1/2)*n, 3/2, (1/2)*x^2)*x
f := x*(diff(y(x), x, x, x))+x*(diff(y(x), x))+y(x)-1
dsolve(f);
y(x) = x*BesselJ(1, x)*_C3+x*BesselY(1, x)*_C2+(1/4)*π*x*(-BesselJ(1, x)*BesselY(0, x)+BesselY(1, x)*BesselJ(0, x))*(Pi*x*_C1*StruveH(-1, x)-2)
y(x) = exp(-(1/4)*x^4)*HeunB(-1/2, 0, -3/2, 1, -(1/2)*x^2)*_C2+exp(-(1/4)*x^4)*HeunB(1/2, 0, -3/2, 1, -(1/2)*x^2)*x*_C1+1
f := diff(y(x), x, x)+(1-x^2)*(diff(y(x), x))+x*y(x)-1
dsolve(f);
y(x) = HeunT(0, 6, -3^(1/3), (1/3)*3^(2/3)*x)*_C2+HeunT(0, -6, -3^(1/3), -(1/3)*3^(2/3)*x)*exp((1/3)*x*(-3+x^2))*_C1+x;
f := diff(y(x), x, x)-cos(x)*y(x)-1
y(x) = MathieuC(0, 2, (1/2)x| _C2 + MathieuS(0, 2, (1/2)x) _C1 - 2 | |
/ / 1 \\// / 1 \ /
|MathieuS|0, 2, - x|| |MathieuC|0, 2, - x| MathieuSPrime|0, 2,
\ \ 2 // \ \ 2 / \
\
1 \ / 1 \ / 1 \\ |
- x| - MathieuS|0, 2, - x| MathieuCPrime|0, 2, - x|| dx|
2 / \ 2 / \ 2 // |
/
/ /
/ 1 \ | | / / 1 \\// /
MathieuC|0, 2, - x| + 2 | | |MathieuC|0, 2, - x|| |MathieuC|0,
\ 2 / | | \ \ 2 // \ \
\/
1 \ / 1 \
2, - x| MathieuSPrime|0, 2, - x|
2 / \ 2 /
\
/ 1 \ / 1 \\ | /
- MathieuS|0, 2, - x| MathieuCPrime|0, 2, - x|| dx| MathieuS|
\ 2 / \ 2 // | \
/
1 \
0, 2, - x|
2 /
>