2.1.12 Problem 12

Internal problem ID [8723]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 12
Date solved : Friday, April 25, 2025 at 04:58:01 PM
CAS classification : [_separable]

Existence and uniqueness analysis

Solve

$$\begin{array}{rl}{y}^{\prime }& =\frac{2y}{x}\end{array}$$

With initial conditions

$$\begin{array}{rl}y\left(0\right)& =0\end{array}$$

This is a linear ODE. In canonical form it is written as

$$\begin{array}{rl}{y}^{\prime }+q(x)y& =p(x)\end{array}$$

Where here

$$\begin{array}{rl}q(x)& =-\frac{2}{x}\\ p(x)& =0\end{array}$$

Hence the ode is

$$\begin{array}{r}{y}^{\prime }-\frac{2y}{x}=0\end{array}$$

The domain of $q(x)=-\frac{2}{x}$ is

$$\{x<0\mathbf{\vee }0<x\}$$

But the point $x_0=0$ is not inside this domain. Hence existence and uniqueness theorem does not apply. There could be infinite number of solutions, or one solution or no solution at all.

Solved using first_order_ode_linear

Time used: 0.034 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime }& =\frac{2y}{x}\end{array}$$

With initial conditions

$$\begin{array}{rl}y\left(0\right)& =0\end{array}$$

In canonical form a linear first order is

$$\begin{array}{rl}{y}^{\prime }+q(x)y& =p(x)\end{array}$$

Comparing the above to the given ode shows that

$$\begin{array}{rl}q(x)& =-\frac{2}{x}\\ p(x)& =0\end{array}$$

The integrating factor $\mu$ is

$$\begin{array}{rl}\mu & =e^{\int qdx}\\ & ={\mathrm{e}}^{\int -\frac{2}{x}dx}\\ & =\frac{1}{x^2}\end{array}$$

The ode becomes

$$\begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}x}\mu y& =0\\ \frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{y}{x^2}\right)& =0\end{array}$$

Integrating gives

$$\begin{array}{rl}\frac{y}{x^2}& =\int 0dx+c_1\\ & =c_1\end{array}$$

Dividing throughout by the integrating factor $\frac{1}{x^2}$ gives the final solution

$$y=c_1{x}^2$$

Solving for the constant of integration from initial conditions, the solution becomes

$$\begin{array}{r}y=0\end{array}$$

Solution plot	Slope field $y^{\prime }=\frac{2y}{x}$

Summary of solutions found

$$\begin{array}{rl}y& =0\end{array}$$

Solved using first_order_ode_separable

Time used: 0.053 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime }& =\frac{2y}{x}\end{array}$$

With initial conditions

$$\begin{array}{rl}y\left(0\right)& =0\end{array}$$

The ode

$$\begin{array}{}\text{(1)}& y^{\prime }=\frac{2y}{x}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}{y}^{\prime }& =\frac{2y}{x}\\ & =f(x)g(y)\end{array}$$

Where

$$\begin{array}{rl}f(x)& =\frac{2}{x}\\ g(y)& =y\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(y)}dy& =\int f(x)dx\\ \int \frac{1}{y}dy& =\int \frac{2}{x}dx\end{array}$$

$$\ln \left(y\right)=\ln \left(x^2\right)+c_2$$

Taking the exponential of both sides the solution becomes

$$y=c_2{x}^2$$

We now need to find the singular solutions, these are found by finding for what values $g(y)$ is zero, since we had to divide by this above. Solving $g(y)=0$ or

$$y=0$$

for $y$ gives

$$\begin{array}{rl}y& =0\end{array}$$

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

$$\begin{array}{rl}y& =0\\ y& =c_2{x}^2\end{array}$$

Solving for the constant of integration from initial conditions, the solution becomes

$$\begin{array}{r}y=0\end{array}$$

Solution plot	Slope field $y^{\prime }=\frac{2y}{x}$

Summary of solutions found

$$\begin{array}{rl}y& =0\end{array}$$

Solved using first_order_ode_homog_A

Time used: 0.193 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime }& =\frac{2y}{x}\end{array}$$

With initial conditions

$$\begin{array}{rl}y\left(0\right)& =0\end{array}$$

In canonical form, the ODE is

$$\begin{array}{rl}{y}^{\prime }& =F(x,y)\\ \text{(1)}& & =\frac{2y}{x}\end{array}$$

An ode of the form $y^{\prime }=\frac{M(x,y)}{N(x,y)}$ is called homogeneous if the functions $M(x,y)$ and $N(x,y)$ are both homogeneous functions and of the same order. Recall that a function $f(x,y)$ is homogeneous of order $n$ if

$$f(t^{n}x,t^{n}y)=t^{n}f(x,y)$$

In this case, it can be seen that both $M=2y$ and $N=x$ are both homogeneous and of the same order $n=1$. Therefore this is a homogeneous ode. Since this ode is homogeneous, it is converted to separable ODE using the substitution $u=\frac{y}{x}$, or $y=ux$. Hence

$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}u}{\mathrm{d}x}x+u$$

Applying the transformation $y=ux$ to the above ODE in (1) gives

$$\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}x}x+u& =2u\\ \frac{\mathrm{d}u}{\mathrm{d}x}& =\frac{u\left(x\right)}{x}\end{array}$$

Or

$$u^{\prime }\left(x\right)-\frac{u\left(x\right)}{x}=0$$

Or

$$u^{\prime }\left(x\right)x-u\left(x\right)=0$$

Which is now solved as separable in $u\left(x\right)$.

The ode

$$\begin{array}{}\text{(2)}& u^{\prime }\left(x\right)=\frac{u\left(x\right)}{x}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}{u}^{\prime }\left(x\right)& =\frac{u\left(x\right)}{x}\\ & =f(x)g(u)\end{array}$$

Where

$$\begin{array}{rl}f(x)& =\frac{1}{x}\\ g(u)& =u\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(u)}du& =\int f(x)dx\\ \int \frac{1}{u}du& =\int \frac{1}{x}dx\end{array}$$

$$\ln \left(u\left(x\right)\right)=\ln \left(x\right)+c_3$$

Taking the exponential of both sides the solution becomes

$$u\left(x\right)=c_{3}x$$

We now need to find the singular solutions, these are found by finding for what values $g(u)$ is zero, since we had to divide by this above. Solving $g(u)=0$ or

$$u=0$$

for $u\left(x\right)$ gives

$$\begin{array}{rl}u\left(x\right)& =0\end{array}$$

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

$$\begin{array}{rl}u\left(x\right)& =0\\ u\left(x\right)& =c_{3}x\end{array}$$

Converting $u\left(x\right)=0$ back to $y$ gives

$$\begin{array}{r}y=0\end{array}$$

Converting $u\left(x\right)=c_{3}x$ back to $y$ gives

$$\begin{array}{r}y=c_3{x}^2\end{array}$$

Solving for the constant of integration from initial conditions, the solution becomes

$$\begin{array}{r}y=0\end{array}$$

Solution plot	Slope field $y^{\prime }=\frac{2y}{x}$

Summary of solutions found

$$\begin{array}{rl}y& =0\end{array}$$

Solved using first_order_ode_homog_type_D2

Time used: 0.047 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime }& =\frac{2y}{x}\end{array}$$

With initial conditions

$$\begin{array}{rl}y\left(0\right)& =0\end{array}$$

Applying change of variables $y=u\left(x\right)x$, then the ode becomes

$$\begin{array}{r}{u}^{\prime }\left(x\right)x+u\left(x\right)=2u\left(x\right)\end{array}$$

Which is now solved The ode

$$\begin{array}{}\text{(3)}& u^{\prime }\left(x\right)=\frac{u\left(x\right)}{x}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}{u}^{\prime }\left(x\right)& =\frac{u\left(x\right)}{x}\\ & =f(x)g(u)\end{array}$$

Where

$$\begin{array}{rl}f(x)& =\frac{1}{x}\\ g(u)& =u\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(u)}du& =\int f(x)dx\\ \int \frac{1}{u}du& =\int \frac{1}{x}dx\end{array}$$

$$\ln \left(u\left(x\right)\right)=\ln \left(x\right)+c_4$$

Taking the exponential of both sides the solution becomes

$$u\left(x\right)=c_{4}x$$

We now need to find the singular solutions, these are found by finding for what values $g(u)$ is zero, since we had to divide by this above. Solving $g(u)=0$ or

$$u=0$$

for $u\left(x\right)$ gives

$$\begin{array}{rl}u\left(x\right)& =0\end{array}$$

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

$$\begin{array}{rl}u\left(x\right)& =0\\ u\left(x\right)& =c_{4}x\end{array}$$

Converting $u\left(x\right)=0$ back to $y$ gives

$$\begin{array}{r}y=0\end{array}$$

Converting $u\left(x\right)=c_{4}x$ back to $y$ gives

$$\begin{array}{r}y=c_4{x}^2\end{array}$$

Solving for the constant of integration from initial conditions, the solution becomes

$$\begin{array}{r}y=0\end{array}$$

Solution plot	Slope field $y^{\prime }=\frac{2y}{x}$

Summary of solutions found

$$\begin{array}{rl}y& =0\end{array}$$

Solved using first_order_ode_homog_type_maple_C

Time used: 0.247 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime }& =\frac{2y}{x}\end{array}$$

With initial conditions

$$\begin{array}{rl}y\left(0\right)& =0\end{array}$$

Let $Y=y-y_0$ and $X=x-x_0$ then the above is transformed to new ode in $Y(X)$

$$\frac{d}{dX}Y\left(X\right)=\frac{2Y\left(X\right)+2y_0}{X+x_0}$$

Solving for possible values of $x_0$ and $y_0$ which makes the above ode a homogeneous ode results in

$$\begin{array}{rl}{x}_0& =0\\ y_0& =0\end{array}$$

Using these values now it is possible to easily solve for $Y\left(X\right)$. The above ode now becomes

$$\begin{array}{r}\frac{d}{dX}Y\left(X\right)=\frac{2Y\left(X\right)}{X}\end{array}$$

In canonical form, the ODE is

$$\begin{array}{rl}{Y}^{\prime }& =F(X,Y)\\ \text{(1)}& & =\frac{2Y}{X}\end{array}$$

An ode of the form $Y^{\prime }=\frac{M(X,Y)}{N(X,Y)}$ is called homogeneous if the functions $M(X,Y)$ and $N(X,Y)$ are both homogeneous functions and of the same order. Recall that a function $f(X,Y)$ is homogeneous of order $n$ if

$$f(t^{n}X,t^{n}Y)=t^{n}f(X,Y)$$

In this case, it can be seen that both $M=2Y$ and $N=X$ are both homogeneous and of the same order $n=1$. Therefore this is a homogeneous ode. Since this ode is homogeneous, it is converted to separable ODE using the substitution $u=\frac{Y}{X}$, or $Y=uX$. Hence

$$\frac{\mathrm{d}Y}{\mathrm{d}X}=\frac{\mathrm{d}u}{\mathrm{d}X}X+u$$

Applying the transformation $Y=uX$ to the above ODE in (1) gives

$$\begin{array}{rl}\frac{\mathrm{d}u}{\mathrm{d}X}X+u& =2u\\ \frac{\mathrm{d}u}{\mathrm{d}X}& =\frac{u\left(X\right)}{X}\end{array}$$

Or

$$\frac{d}{dX}u\left(X\right)-\frac{u\left(X\right)}{X}=0$$

Or

$$\left(\frac{d}{dX}u\left(X\right)\right)X-u\left(X\right)=0$$

Which is now solved as separable in $u\left(X\right)$.

The ode

$$\begin{array}{}\text{(4)}& \frac{d}{dX}u\left(X\right)=\frac{u\left(X\right)}{X}\end{array}$$

is separable as it can be written as

$$\begin{array}{rl}\frac{d}{dX}u\left(X\right)& =\frac{u\left(X\right)}{X}\\ & =f(X)g(u)\end{array}$$

Where

$$\begin{array}{rl}f(X)& =\frac{1}{X}\\ g(u)& =u\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{g(u)}du& =\int f(X)dX\\ \int \frac{1}{u}du& =\int \frac{1}{X}dX\end{array}$$

$$\ln \left(u\left(X\right)\right)=\ln \left(X\right)+c_5$$

Taking the exponential of both sides the solution becomes

$$u\left(X\right)=c_{5}X$$

We now need to find the singular solutions, these are found by finding for what values $g(u)$ is zero, since we had to divide by this above. Solving $g(u)=0$ or

$$u=0$$

for $u\left(X\right)$ gives

$$\begin{array}{rl}u\left(X\right)& =0\end{array}$$

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

Therefore the solutions found are

$$\begin{array}{rl}u\left(X\right)& =0\\ u\left(X\right)& =c_{5}X\end{array}$$

Converting $u\left(X\right)=0$ back to $Y\left(X\right)$ gives

$$\begin{array}{r}Y\left(X\right)=0\end{array}$$

Converting $u\left(X\right)=c_{5}X$ back to $Y\left(X\right)$ gives

$$\begin{array}{r}Y\left(X\right)=X^2{c}_5\end{array}$$

Using the solution for $Y(X)$

$$\begin{array}{}\text{(A)}& Y\left(X\right)=0\end{array}$$

And replacing back terms in the above solution using

$$\begin{array}{rl}Y& =y+y_0\\ X& =x_0+x\end{array}$$

Or

$$\begin{array}{rl}Y& =y\\ X& =x\end{array}$$

Then the solution in $y$ becomes using EQ (A)

$$\begin{array}{r}y=0\end{array}$$

Using the solution for $Y(X)$

$$\begin{array}{}\text{(A)}& Y\left(X\right)=X^2{c}_5\end{array}$$

And replacing back terms in the above solution using

$$\begin{array}{rl}Y& =y+y_0\\ X& =x_0+x\end{array}$$

Or

$$\begin{array}{rl}Y& =y\\ X& =x\end{array}$$

Then the solution in $y$ becomes using EQ (A)

$$\begin{array}{r}y=x^2{c}_5\end{array}$$

Solving for the constant of integration from initial conditions, the solution becomes

$$\begin{array}{r}y=0\end{array}$$

Solution plot	Slope field $y^{\prime }=\frac{2y}{x}$

Summary of solutions found

$$\begin{array}{rl}y& =0\end{array}$$

Solved using first_order_ode_exact

Time used: 0.077 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime }& =\frac{2y}{x}\end{array}$$

With initial conditions

$$\begin{array}{rl}y\left(0\right)& =0\end{array}$$

To solve an ode of the form

$$\begin{array}{}\text{(A)}& M(x,y)+N(x,y)\frac{dy}{dx}=0\end{array}$$

We assume there exists a function $\varphi (x,y)=c$ where $c$ is constant, that satisfies the ode. Taking derivative of $\varphi$ w.r.t. $x$ gives

$$\frac{d}{dx}\varphi (x,y)=0$$

Hence

$$\begin{array}{}\text{(B)}& \frac{\partial \varphi }{\partial x}+\frac{\partial \varphi }{\partial y}\frac{dy}{dx}=0\end{array}$$

Comparing (A,B) shows that

$$\begin{array}{rl}\frac{\partial \varphi }{\partial x}& =M\\ \frac{\partial \varphi }{\partial y}& =N\end{array}$$

But since $\frac{{\partial }^2\varphi }{\partial x\partial y}=\frac{{\partial }^2\varphi }{\partial y\partial x}$ then for the above to be valid, we require that

$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$

If the above condition is satisfied, then the original ode is called exact. We still need to determine $\varphi (x,y)$ but at least we know now that we can do that since the condition $\frac{{\partial }^2\varphi }{\partial x\partial y}=\frac{{\partial }^2\varphi }{\partial y\partial x}$ is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. The first step is to write the ODE in standard form to check for exactness, which is

$$\begin{array}{}\text{(1A)}& M(x,y)\mathrm{d}x+N(x,y)\mathrm{d}y=0\end{array}$$

Therefore

$$\begin{array}{rl}\mathrm{d}y& =\left(\frac{2y}{x}\right)\mathrm{d}x\\ \text{(2A)}& (-\frac{2y}{x})\mathrm{d}x+\mathrm{d}y& =0\end{array}$$

Comparing (1A) and (2A) shows that

$$\begin{array}{rl}M(x,y)& =-\frac{2y}{x}\\ N(x,y)& =1\end{array}$$

The next step is to determine if the ODE is is exact or not. The ODE is exact when the following condition is satisfied

$$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$$

Using result found above gives

$$\begin{array}{rl}\frac{\partial M}{\partial y}& =\frac{\partial }{\partial y}(-\frac{2y}{x})\\ & =-\frac{2}{x}\end{array}$$

And

$$\begin{array}{rl}\frac{\partial N}{\partial x}& =\frac{\partial }{\partial x}\left(1\right)\\ & =0\end{array}$$

Since $\frac{\partial M}{\partial y}\ne \frac{\partial N}{\partial x}$, then the ODE is not exact. Since the ODE is not exact, we will try to find an integrating factor to make it exact. Let

$$\begin{array}{rl}A& =\frac{1}{N}(\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x})\\ & =1((-\frac{2}{x})-\left(0\right))\\ & =-\frac{2}{x}\end{array}$$

Since $A$ does not depend on $y$, then it can be used to find an integrating factor. The integrating factor $\mu$ is

$$\begin{array}{rl}\mu & =e^{\int A\mathrm{d}x}\\ & =e^{\int -\frac{2}{x}\mathrm{d}x}\end{array}$$

The result of integrating gives

$$\begin{array}{rl}\mu & =e^{-2\ln \left(x\right)}\\ & =\frac{1}{x^2}\end{array}$$

$M$ and $N$ are multiplied by this integrating factor, giving new $M$ and new $N$ which are called $\bar{M}$ and $\bar{N}$ for now so not to confuse them with the original $M$ and $N$.

$$\begin{array}{rl}\bar{M}& =\mu M\\ & =\frac{1}{x^2}(-\frac{2y}{x})\\ & =-\frac{2y}{x^3}\end{array}$$

And

$$\begin{array}{rl}\bar{N}& =\mu N\\ & =\frac{1}{x^2}\left(1\right)\\ & =\frac{1}{x^2}\end{array}$$

Now a modified ODE is ontained from the original ODE, which is exact and can be solved. The modified ODE is

$$\begin{array}{rl}\bar{M}+\bar{N}\frac{\mathrm{d}y}{\mathrm{d}x}& =0\\ (-\frac{2y}{x^3})+\left(\frac{1}{x^2}\right)\frac{\mathrm{d}y}{\mathrm{d}x}& =0\end{array}$$

The following equations are now set up to solve for the function $\varphi (x,y)$

$$\begin{array}{}\text{(1)}& \frac{\partial \varphi }{\partial x}& =\bar{M}\text{(2)}& \frac{\partial \varphi }{\partial y}& =\bar{N}\end{array}$$

Integrating (2) w.r.t. $y$ gives

$$\begin{array}{rl}\int \frac{\partial \varphi }{\partial y}\mathrm{d}y& =\int \bar{N}\mathrm{d}y\\ \int \frac{\partial \varphi }{\partial y}\mathrm{d}y& =\int \frac{1}{x^2}\mathrm{d}y\\ \text{(3)}& \varphi & =\frac{y}{x^2}+f(x)\end{array}$$

Where $f(x)$ is used for the constant of integration since $\varphi$ is a function of both $x$ and $y$. Taking derivative of equation (3) w.r.t $x$ gives

$$\begin{array}{}\text{(4)}& \frac{\partial \varphi }{\partial x}=-\frac{2y}{x^3}+f^{\prime }(x)\end{array}$$

But equation (1) says that $\frac{\partial \varphi }{\partial x}=-\frac{2y}{x^3}$. Therefore equation (4) becomes

$$\begin{array}{}\text{(5)}& -\frac{2y}{x^3}=-\frac{2y}{x^3}+f^{\prime }(x)\end{array}$$

Solving equation (5) for $f^{\prime }(x)$ gives

$$f^{\prime }(x)=0$$

Therefore

$$f(x)=c_6$$

Where $c_6$ is constant of integration. Substituting this result for $f(x)$ into equation (3) gives $\varphi$

$$\varphi =\frac{y}{x^2}+c_6$$

But since $\varphi$ itself is a constant function, then let $\varphi =c_7$ where $c_7$ is new constant and combining $c_6$ and $c_7$ constants into the constant $c_6$ gives the solution as

$$c_6=\frac{y}{x^2}$$

Solving for the constant of integration from initial conditions, the solution becomes

$$\begin{array}{r}\frac{y}{x^2}=0\end{array}$$

Solving for $y$ gives

$$\begin{array}{rl}y& =0\end{array}$$

Solution plot	Slope field $y^{\prime }=\frac{2y}{x}$

Summary of solutions found

$$\begin{array}{rl}y& =0\end{array}$$

✓ Maple. Time used: 0.039 (sec). Leaf size: 9

ode:=diff(y(x),x) = 2*y(x)/x; 
ic:=y(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

$$y=c_1{x}^2$$

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & [\frac{d}{dx}y\left(x\right)=\frac{2y\left(x\right)}{x},y\left(0\right)=0]\\ \bullet & & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{d}{dx}y\left(x\right)\\ \bullet & & \text{Solve for the highest derivative}\\ & & \frac{d}{dx}y\left(x\right)=\frac{2y\left(x\right)}{x}\\ \bullet & & \text{Separate variables}\\ & & \frac{\frac{d}{dx}y\left(x\right)}{y\left(x\right)}=\frac{2}{x}\\ \bullet & & \text{Integrate both sides with respect to}x\\ & & \int \frac{\frac{d}{dx}y\left(x\right)}{y\left(x\right)}dx=\int \frac{2}{x}dx+\mathit{C}\mathit{1}\\ \bullet & & \text{Evaluate integral}\\ & & \ln \left(y\left(x\right)\right)=2\ln \left(x\right)+\mathit{C}\mathit{1}\\ \bullet & & \text{Solve for}y\left(x\right)\\ & & y\left(x\right)={\mathrm{e}}^{\mathit{C}\mathit{1}}{x}^2\\ \bullet & & \text{Redefine the integration constant(s)}\\ & & y\left(x\right)=\mathit{C}\mathit{1}{x}^2\\ \bullet & & \text{Use initial condition}y\left(0\right)=0\\ & & 0=0\\ \bullet & & \text{Solve for}\mathit{\text{\_C1}}\\ & & \mathit{C}\mathit{1}=\mathit{C}\mathit{1}\\ \bullet & & \text{Substitute}\mathit{\text{\_C1}}=\mathit{\text{\_C1}}\text{into general solution and simplify}\\ & & y\left(x\right)=\mathit{C}\mathit{1}{x}^2\\ \bullet & & \text{Solution to the IVP}\\ & & y\left(x\right)=\mathit{C}\mathit{1}{x}^2\end{array}$$

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 6

ode=D[y[x],x] == 2*y[x]/x; 
ic=y[0]==0; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$y(x)\to 0$$

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - 2*y(x)/x,0) 
ics = {y(0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Couldnt solve for initial conditions