General Solution of ODE with Arbitrary Constant Coefficient

Question

The general solution of ${x^2}y'' + 3xy' + \lambda y = 0$ is (from Example 4 here) $$y(x) = \begin{cases} {c_1}{x^{ - 1}}\cos \left( {\ln \left( x \right)\sqrt {\lambda - 1} } \right) + {c_2}{x^{ - 1}}\sin \left( {\ln \left( x \right)\sqrt {\lambda - 1} } \right) &\mbox{for } \lambda > 1 \\ {c_1}{x^{ - 1}} + {c_2}{x^{ - 1}}\ln \left( x \right) &\mbox{for } \lambda = 1\\ {c_1}{x^{ - 1 - \sqrt {1 - \lambda } }} + {c_2}{x^{ - 1 + \sqrt {1 - \lambda } }} &\mbox{for } \lambda < 1 \end{cases}$$

Why does Mathematica give only the third case instead of the whole solution? Is it a bug?

FullSimplify[DSolve[x^2*y''[x] + 3*x*y'[x] + \[Lambda]*y[x] == 0, y[x], x], Assumptions -> \[Lambda] \[Element] Reals]

Answer 1

Mathematica gave generic solution to Euler ode.

Using assumptions and little bit of known manipulation, you can obtain the solutions given on that web page

For $\lambda>1$

Clear["Global`*"];
ode = x^2*y''[x] + 3*x*y'[x] + lam*y[x] == 0;
sol = y[x] /. First@DSolve[ode, y[x], x, Assumptions -> lam > 1]

The only trick is to convert the above to trig, using Euler relations. There might be easier way to do this, but I could not find it now. We need to implement the following transformation

\begin{align*} c_1 x^{\alpha + i \beta}+ c_1 x^{\alpha - i \beta}&= x^\alpha (c_1 x^{i \beta}+c_2 x^{ -i \beta})\\ &=x^\alpha(c_1 e^{\ln x^{i \beta}}+c_2 e^{\ln x^{-i \beta}}\\ &=x^\alpha(c_1 e^{i \beta \ln x}+c_2 e^{-i \beta \ln x}\\ &=x^\alpha(c_1 \cos(\beta \ln x)+ c_2 \sin(\beta \ln x)\\ \end{align*}

The last step above is just Euler's relation.

The above is done using:

ClearAll[a, b];
exponent = sol[[1]] /. Times[Power[x, a_], any_] :> Expand[a];
{a, b} = exponent /. a_ - I b_ :> {Simplify@a, Simplify@b};
sol = x^a (C[1]*Cos[b*Log[x]] + C[2] Sin[b*Log[x]])

For the other two cases, no need to help Mathematica, it gives the solutions as is, when using assumptions

$\lambda =1 $

  sol = y[x] /. First@DSolve[ode /. lam -> 1, y[x], x]

$\lambda < 1 $

  sol = y[x] /. First@DSolve[ode, y[x], x, Assumptions -> lam < 1] 

Update to answer comment

What do you mean by "generic solution to Euler ode" in the first line?

I mean the following. This is what Mathematica basically did

$$ x^{2}y^{\prime\prime}+3xy^{\prime}+\lambda y=0 $$

Let $y=Cx^{r}$. Substituting into the ODE this gives \begin{align*} x^{2}Cr\left( r-1\right) x^{r-2}+3xCrx^{r-1}+\lambda Ax^{r} & =0\\ Cr\left( r-1\right) x^{r}+3Crx^{r}+\lambda Cx^{r} & =0 \end{align*}

Simplifying, since $C x^{r}\neq0$ gives

\begin{align*} r\left( r-1\right) +3r+\lambda & =0\\ r^{2}-r+3r+\lambda & =0\\ r^{2}+2r+\lambda & =0 \end{align*}

Using the Quadratic formula, $r=\frac{-b}{2a}\pm\frac{1}{2a}\sqrt{b^{2} -4ac}=\frac{-2}{2}\pm\frac{1}{2}\sqrt{4-4\lambda}=-1\pm\sqrt{1-\lambda}$. Therefore

\begin{align*} r_{1} & =-1+\sqrt{1-\lambda}\\ r_{2} & =-1-\sqrt{1-\lambda} \end{align*}

Hence the general solution is the sum of the two basis solutions given by

\begin{align*} y & =C_{1}x^{r_{1}}+C_{2}x^{r_{2}}\\ & =C_{1}x^{-1+\sqrt{1-\lambda}}+C_{2}x^{-1-\sqrt{1-\lambda}} \end{align*}

It is a generic solution, since it does not know anything about $\lambda$ it could not simplify this any more.

DSolve[x^2*y''[x] + 3*x*y'[x] + \[Lambda]*y[x] == 0, y[x], x] // 
  Simplify // Expand

Answer 2

Here's a way to generate any solution, by picking a different basis and taking the limit as the parameter approaches a given value of $\lambda$:

ode = x^2*y''[x] + 3*x*y'[x] + λ*y[x] == 0;
y2 = DSolveValue[{ode, y[1] == 0, y'[1] == 1}, y[x], x];
y1 = DSolveValue[{ode, y[1] == 1, y'[1] == -1}, y[x], x];
ClearAll[solg];
solg[λ0_?NumericQ] := 
 Assuming[x > 0, 
  Limit[C[1] y1 + C[2] y2, λ -> λ0] // Re // 
    ComplexExpand // Simplify]

Examples:

solg[1]
ode /. {λ -> 1, 
   y -> Function[x, Evaluate[solg[1]]]} // Simplify
(*
  (C[1] + C[2] Log[x])/x
  True
*)

solg[2]
ode /. {λ -> 2, 
   y -> Function[x, Evaluate[solg[2]]]} // Simplify
(*
  (C[1] Cos[Log[x]] + C[2] Sin[Log[x]])/x
  True
*)

solg[0]
ode /. {λ -> 0, 
   y -> Function[x, Evaluate[solg[0]]]} // Simplify
(*
  ((1 + x^2) C[1] + (-1 + x^2) C[2])/(2 x^2)
  True
*)