Michael Gustafson

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I was looking at symbolically solving a second-order differential equation and it looks like the method=laplace method has a sign error when the coefficients are presented in a certain way.  Below is a picture of some examples with and without method=laplace that should all have the same closed form.  Note that lines (s6) and (s8) have different signs in the exponential than they should have (which is a HUGE problem):

restart

s1 := dsolve([diff(x(t), t, t)+2*a*(diff(x(t), t))+a^2*x(t)], [x(t)])

{x(t) = exp(-a*t)*(_C2*t+_C1)}

(1)

s2 := dsolve([diff(x(t), t, t)+2*a*(diff(x(t), t))+a^2*x(t)], [x(t)], method = laplace)

x(t) = exp(-a*t)*(t*(D(x))(0)+x(0)*(a*t+1))

(2)

s3 := dsolve([diff(x(t), t, t)+2*(diff(x(t), t))/b+x(t)/b^2], [x(t)])

{x(t) = exp(-t/b)*(_C2*t+_C1)}

(3)

s4 := dsolve([diff(x(t), t, t)+2*(diff(x(t), t))/b+x(t)/b^2], [x(t)], method = laplace)

x(t) = exp(-t/b)*(t*(D(x))(0)+x(0)*(b+t)/b)

(4)

s5 := dsolve([diff(x(t), t, t)+2*(diff(x(t), t))/sqrt(L*C)+x(t)/(L*C)], [x(t)])

{x(t) = exp(-(L*C)^(1/2)*t/(L*C))*(_C2*t+_C1)}

(5)

s6 := dsolve([diff(x(t), t, t)+2*(diff(x(t), t))/sqrt(L*C)+x(t)/(L*C)], [x(t)], method = laplace)

x(t) = (t*(D(x))(0)+3*C*L*x(0)*t/(L*C)^(3/2)+x(0))*exp((L*C)^(1/2)*t/(L*C))

(6)

s7 := dsolve([L*C*(diff(x(t), t, t))+2*sqrt(L*C)*(diff(x(t), t))+x(t)], [x(t)])

{x(t) = exp(-(L*C)^(1/2)*t/(L*C))*(_C2*t+_C1)}

(7)

s8 := dsolve([L*C*(diff(x(t), t, t))+2*sqrt(L*C)*(diff(x(t), t))+x(t)], [x(t)], method = laplace)

x(t) = exp(t/(L*C)^(1/2))*(t*(D(x))(0)+x(0)*(L*C+3*(L*C)^(1/2)*t)/(L*C))

(8)

s9 := dsolve([diff(x(t), t, t)+2*z*wn*(diff(x(t), t))+wn^2*x(t)], [x(t)])

{x(t) = _C1*exp((-z+(z^2-1)^(1/2))*wn*t)+_C2*exp(-(z+(z^2-1)^(1/2))*wn*t)}

(9)

s10 := dsolve([diff(x(t), t, t)+2*z*wn*(diff(x(t), t))+wn^2*x(t)], [x(t)], method = laplace)

x(t) = exp(-wn*t*z)*(cosh((wn^2*(z^2-1))^(1/2)*t)*x(0)+(x(0)*wn*z+(D(x))(0))*sinh((wn^2*(z^2-1))^(1/2)*t)/(wn^2*(z^2-1))^(1/2))

(10)

s11 := dsolve([(diff(x(t), t, t))/wn^2+2*z*(diff(x(t), t))/wn+x(t)], [x(t)])

{x(t) = _C1*exp((-z+(z^2-1)^(1/2))*wn*t)+_C2*exp(-(z+(z^2-1)^(1/2))*wn*t)}

(11)

s12 := dsolve([(diff(x(t), t, t))/wn^2+2*z*(diff(x(t), t))/wn+x(t)], [x(t)], method = laplace)

x(t) = exp(-wn*t*z)*(cosh((wn^2*(z^2-1))^(1/2)*t)*x(0)+(x(0)*wn*z+(D(x))(0))*sinh((wn^2*(z^2-1))^(1/2)*t)/(wn^2*(z^2-1))^(1/2))

(12)

s13 := dsolve([(diff(x(t), t, t))/wn^2+2*z*(diff(x(t), t))/wn+x(t)], [x(t)])

{x(t) = _C1*exp((-z+(z^2-1)^(1/2))*wn*t)+_C2*exp(-(z+(z^2-1)^(1/2))*wn*t)}

(13)

s14 := dsolve([(diff(x(t), t, t))/wn^2+2*z*(diff(x(t), t))/wn+x(t)], [x(t)], method = laplace)

x(t) = exp(-wn*t*z)*(cosh((wn^2*(z^2-1))^(1/2)*t)*x(0)+(x(0)*wn*z+(D(x))(0))*sinh((wn^2*(z^2-1))^(1/2)*t)/(wn^2*(z^2-1))^(1/2))

(14)

NULL

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