2.1.70 Problem 70

Internal problem ID [8782]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 70
Date solved : Sunday, March 30, 2025 at 01:35:41 PM
CAS classification : [[_2nd_order, _quadrature]]

Solved as second order missing x ode

Time used: 0.530 (sec)

Solve

$$\begin{array}{rl}ay{y}^{\prime \prime }+by& =0\end{array}$$

This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable $y$ an independent variable. Using

$$\begin{array}{rl}{y}^{\prime }& =p\end{array}$$

Then

$$\begin{array}{rl}{y}^{″}& =\frac{dp}{dx}\\ & =\frac{dp}{dy}\frac{dy}{dx}\\ & =p\frac{dp}{dy}\end{array}$$

Hence the ode becomes

$$\begin{array}{r}ayp\left(y\right)\left(\frac{d}{dy}p\left(y\right)\right)+by=0\end{array}$$

Which is now solved as first order ode for $p(y)$.

Integrating gives

$$\begin{array}{rl}\int -\frac{pa}{b}dp& =dy\\ -\frac{p^{2}a}{2b}& =y+c_1\end{array}$$

For solution (1) found earlier, since $p=y^{\prime }$ then we now have a new first order ode to solve which is

$$\begin{array}{r}-\frac{{y^{\prime }}^{2}a}{2b}=y+c_1\end{array}$$

Let $p=y^{\prime }$ the ode becomes

$$\begin{array}{r}-\frac{p^{2}a}{2b}=y+c_1\end{array}$$

Solving for $y$ from the above results in

$$\begin{array}{}\text{(1)}& y& =-\frac{p^{2}a+2c_{1}b}{2b}\end{array}$$

This has the form

$$\begin{array}{}\text{(*)}& y=xf(p)+g(p)\end{array}$$

Where $f,g$ are functions of $p=y^{\prime }(x)$. The above ode is dAlembert ode which is now solved.

Taking derivative of (*) w.r.t. $x$ gives

$$\begin{array}{rl}p& =f+(x{f}^{\prime }+g^{\prime })\frac{dp}{dx}\\ \text{(2)}& p-f& =(x{f}^{\prime }+g^{\prime })\frac{dp}{dx}\end{array}$$

Comparing the form $y=xf+g$ to (1A) shows that

$$\begin{array}{rl}f& =0\\ g& =\frac{-p^{2}a-2c_{1}b}{2b}\end{array}$$

Hence (2) becomes

$$\begin{array}{}\text{(2A)}& p=-\frac{ap{p}^{\prime }\left(x\right)}{b}\end{array}$$

The singular solution is found by setting $\frac{dp}{dx}=0$ in the above which gives

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

Solving the above for $p$ results in

$$\begin{array}{rl}{p}_1& =0\end{array}$$

Substituting these in (1A) and keeping singular solution that verifies the ode gives

$$\begin{array}{r}y=-c_1\end{array}$$

The general solution is found when $\frac{\mathrm{d}p}{\mathrm{d}x}\ne 0$. From eq. (2A). This results in

$$\begin{array}{}\text{(3)}& p^{\prime }\left(x\right)=-\frac{b}{a}\end{array}$$

This ODE is now solved for $p\left(x\right)$. No inversion is needed.

Since the ode has the form $p^{\prime }\left(x\right)=f(x)$, then we only need to integrate $f(x)$.

$$\begin{array}{rl}\int dp& =\int -\frac{b}{a}dx\\ p\left(x\right)& =-\frac{bx}{a}+c_2\end{array}$$

Substituing the above solution for $p$ in (2A) gives

$$\begin{array}{rl}y& =-\frac{c_2^2{a}^2-2c_{2}abx+b^2{x}^2+2c_{1}ab}{2ab}\end{array}$$

Will add steps showing solving for IC soon.

The solution

$$y=-c_1$$

was found not to satisfy the ode or the IC. Hence it is removed.

Summary of solutions found

$$\begin{array}{rl}y& =-\frac{c_2^2{a}^2-2c_{2}abx+b^2{x}^2+2c_{1}ab}{2ab}\end{array}$$

✓ Maple. Time used: 0.002 (sec). Leaf size: 22

ode:=a*y(x)*diff(diff(y(x),x),x)+b*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

$$\begin{array}{rl}y& =0\\ y& =-\frac{b{x}^2}{2a}+c_{1}x+c_2\end{array}$$

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & ay\left(x\right)\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)+by\left(x\right)=0\\ \bullet & & \text{Highest derivative means the order of the ODE is}2\\ & & \frac{d}{dx}\frac{d}{dx}y\left(x\right)\\ \bullet & & \text{Isolate 2nd derivative}\\ & & \frac{d}{dx}\frac{d}{dx}y\left(x\right)=-\frac{b}{a}\\ \bullet & & \text{Characteristic polynomial of homogeneous ODE}\\ & & r^2=0\\ \bullet & & \text{Use quadratic formula to solve for}r\\ & & r=\frac{0\pm \left(\sqrt{0}\right)}{2}\\ \bullet & & \text{Roots of the characteristic polynomial}\\ & & r=0\\ \bullet & & \text{1st solution of the homogeneous ODE}\\ & & y_1\left(x\right)=1\\ \bullet & & \text{Repeated root, multiply}{y}_1\left(x\right)\text{by}x\text{to ensure linear independence}\\ & & y_2\left(x\right)=x\\ \bullet & & \text{General solution of the ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}{y}_1\left(x\right)+\mathit{C}\mathit{2}{y}_2\left(x\right)+y_p\left(x\right)\\ \bullet & & \text{Substitute in solutions of the homogeneous ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}+\mathit{C}\mathit{2}x+y_p\left(x\right)\\ ◻& & \text{Find a particular solution}{y}_p\left(x\right)\text{of the ODE}\\ & \circ & \text{Use variation of parameters to find}{y}_p\text{here}f\left(x\right)\text{is the forcing function}\\ & & [y_p\left(x\right)=-y_1\left(x\right)\int \frac{y_2\left(x\right)f\left(x\right)}{W(y_1\left(x\right),y_2\left(x\right))}dx+y_2\left(x\right)\int \frac{y_1\left(x\right)f\left(x\right)}{W(y_1\left(x\right),y_2\left(x\right))}dx,f\left(x\right)=-\frac{b}{a}]\\ & \circ & \text{Wronskian of solutions of the homogeneous equation}\\ & & W(y_1\left(x\right),y_2\left(x\right))=\left[\begin{array}{cc}1& x\\ 0& 1\end{array}\right]\\ & \circ & \text{Compute Wronskian}\\ & & W(y_1\left(x\right),y_2\left(x\right))=1\\ & \circ & \text{Substitute functions into equation for}{y}_p\left(x\right)\\ & & y_p\left(x\right)=-\frac{b(\int 1dxx-\int xdx)}{a}\\ & \circ & \text{Compute integrals}\\ & & y_p\left(x\right)=-\frac{b{x}^2}{2a}\\ \bullet & & \text{Substitute particular solution into general solution to ODE}\\ & & y\left(x\right)=\mathit{C}\mathit{1}+\mathit{C}\mathit{2}x-\frac{b{x}^2}{2a}\end{array}$$

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 28

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

$$\begin{array}{r}y(x)\to 0\\ y(x)\to -\frac{b{x}^2}{2a}+c_{2}x+c_1\end{array}$$

✓ Sympy. Time used: 0.214 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*y(x)*Derivative(y(x), (x, 2)) + b*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

$$[y\left(x\right)=C_1+C_{2}x-\frac{b{x}^2}{2a},y\left(x\right)=0]$$