Informative line

### Definition Of Differential Equations And Their Solutions

Learn definition of a Differential Equation & order of differential equations, Practice to find particular solution of the differential equation problems.

# Differential Equation and Its Order

• An equation that contains independent variable (usually $$'x'$$) some functions of $$x$$, dependent variable (usually $$'y'$$) some functions of $$y$$ and at least one derivative term of $$y$$ with respect to  $$x$$ is called a differential equation.

e.g (1)  $$\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}=sin\,x$$

(2)  $$\dfrac{2ydy}{dx}+lnx=sin^2\,x$$

(3)  $${\sqrt{1+\dfrac {dy}{dx}}}=\dfrac{d^2y}{dx^2}$$ are all differential equations.

• The order of differential equations is the order of the highest order derivative that occurs in the equations.

Example: (1)  $$\dfrac{d^2y}{dx^2}+\dfrac{xdy}{dx}=5sin\,x$$ is of order 2. $$\left(\because\,\dfrac{d^2y}{dx^2}\text{represents order 2}\right)$$

(2)  $$x^3\dfrac{dy}{dx}+lnx=5$$ is of order 1. $$\left(\because\,\dfrac{dy}{dx}\text{represents order 1}\right)$$

(3)  $$5y^3\dfrac{d^3y}{dx^3}=2x+\dfrac{dy}{dx}=5sin\,x$$ is of order 3. $$\left(\because\,\dfrac{d^3y}{dx^3}\text{represents order 3}\right)$$

#### Which of the following equations is a differential equation?

A $$2y^3–x^2=sin\,x$$

B $$cos^3x+e^4\,lnx=2$$

C $$5y+x=7$$

D $$\dfrac{xdy}{dx}+y=12cos^3x$$

×

Differential equations should contain at least one derivative term, other than $$'x'$$ and $$'y'$$ terms.

Options 'A', 'B' and 'C' do not contain any derivative term, while 'd' has a $$\dfrac{dy}{dx}$$ term

$$\therefore$$ Option 'D' is correct.

### Which of the following equations is a differential equation?

A

$$2y^3–x^2=sin\,x$$

.

B

$$cos^3x+e^4\,lnx=2$$

C

$$5y+x=7$$

D

$$\dfrac{xdy}{dx}+y=12cos^3x$$

Option D is Correct

#### The order of the differential equation  $$5y^2\dfrac{dy}{dx}+2sin^2x=\dfrac{d^2y}{dx^2}$$ is

A 1

B 2

C 3

D 4

×

The order of differential equation is the order of highest order derivative that occurs in the equations.

$$\dfrac{d^2y}{dx^2}$$ is the highest order derivative in the equations.

$$\therefore$$ Order of differential equation is 2

### The order of the differential equation  $$5y^2\dfrac{dy}{dx}+2sin^2x=\dfrac{d^2y}{dx^2}$$ is

A

1

.

B

2

C

3

D

4

Option B is Correct

#### If $$y=sin\,kx$$ is a solution to the differential equation $$36y''=–49y$$. Then the value of $$'k'$$ is

A $$k=\pm\dfrac{5}{4}$$

B $$k=\pm{3}$$

C $$k=\pm{9}$$

D $$k=\pm\dfrac{7}{6}$$

×

$$y=f(x)$$ is a solution to a differential equation if the equation is satisfied when $$y=f(x)$$ and its derivatives are substituted in the equation.

In this case :

$$y=sin\,kx\Rightarrow y'=\,\dfrac{dy}{dx}=kcos\,kx \Rightarrow y''=\,\dfrac{d^2y}{dx^2}$$

$$=–k^2sin\,kx$$

Putting the above in the differential equation have

$$36×–k^2sin\,kx=–49sin\,kx$$

$$36k^2=49\Rightarrow\,k=\pm\dfrac{7}{6}\Rightarrow\,k=\dfrac{7}{6}$$ or $$k=–\dfrac{7}{6}$$

### If $$y=sin\,kx$$ is a solution to the differential equation $$36y''=–49y$$. Then the value of $$'k'$$ is

A

$$k=\pm\dfrac{5}{4}$$

.

B

$$k=\pm{3}$$

C

$$k=\pm{9}$$

D

$$k=\pm\dfrac{7}{6}$$

Option D is Correct

# General and Particular Solution

When the solution to a differential equation is asked usually all solutions have to be reported

• Consider the differential equation

$$f'(x)=f(x)\to$$ we know that the solution to this equation is $$f(x)=e^x$$ but all function of the form $$f(x)=e^x$$ are also have solutions we say that $$y=ce^x$$ is called the general solution (or family of solution) to the differential equation.

• sometimes we are not interested in finding general solution but a solution that satisfies some particular condition, such a solution is called particular solution to the differential equation. The condition is called initial condition and the problem is called initial value problem (IVP).

e.g.  $$y'=y\to y(0)=1^{ \nearrow^{\text {initial condition}}}$$

$$\Rightarrow\,y=ce^x \to$$ general solution

Use initial condition

$$1=ce^0\Rightarrow\,c=1$$

$$\therefore\,y=e^x \to$$ particular solution.

#### If $$y=xe^x+ce^x$$ is the general solution to the differential equation $$y'-y=e^x$$, find the solution to the initial value problem $$y(0)=1$$.

A $$y=2sinx+lnx$$

B $$y=e^x(x+1)$$

C $$y=ce^x+4$$ (C $$\in$$R)

D $$y=e^x(x^2+1)$$

×

Find 'c' using the initial condition and then put this of c in the general solution to get particular solution.

General solution is $$y=xe^x+ce^x$$

Use $$y(0)=1\Rightarrow\,1=0e^0+ce^0\Rightarrow1=0+c\Rightarrow c=1$$

$$\therefore\,$$ The particular solution or solution to initial value problem is

$$y=xe^x+e^x\Rightarrow\,y=e^x(x+1)$$

### If $$y=xe^x+ce^x$$ is the general solution to the differential equation $$y'-y=e^x$$, find the solution to the initial value problem $$y(0)=1$$.

A

$$y=2sinx+lnx$$

.

B

$$y=e^x(x+1)$$

C

$$y=ce^x+4$$ (C $$\in$$R)

D

$$y=e^x(x^2+1)$$

Option B is Correct

# Solution of a Differential Equation

• A function $$f$$ is called solution to a differential equation if the equation is satisfied when $$y=f(x)$$ and its derivative are substituted in the equation.

$$f(x)$$ is a solution to $$y'=5y$$ if

$$f'(x)=5f(x)$$

• $$f(x)=x^2+2x$$ is a solution to $$y'=2x+2$$
• one differential equation may have more than one solution that's why we have to check all the options one by one rather than finding the particular solution.

#### Which of the following is a solution to the differential equation $$\dfrac{d^2y}{dx^2}=\dfrac{1}{y}\left(\dfrac{dy}{dx}\right)^2$$?

A $$f(x)=y=sin^2x$$

B $$f(x)=y=cos^2x$$

C $$f(x)=y=2e^{3x}$$

D $$f(x)=y=ln2x$$

×

$$y=f(x)$$ is a solution to a differential equation if the equation is satisfied when $$y=f(x)$$ and its derivative is substituted in the equation.

For option (A)

$$y=sin^2x\Rightarrow\,\dfrac{dy}{dx}=2sinx\,cosx=sin2x$$ & $$\dfrac{d^2y}{dx^2}=2cos2x$$

substituting in the differential equation, we have

$$2cos2x=\dfrac{1}{sin^2x}×sin^22x\to$$not true

For option (B)

$$y=cos^2x\Rightarrow\,\dfrac{dy}{dx}=-2cosx\,sinx=-sin2x$$ & $$\dfrac{d^2y}{dx^2}=-2cos2x$$

substituting in the differential equation, we have

$$-2cos2x=\dfrac{1}{cos^2x}×sin^22x\to$$not true

For option (C)

$$y=2e^{3x}\Rightarrow\,\dfrac{dy}{dx}=6e^{3x} \,\& \,\dfrac{d^2y}{dx^2}=18e^{3x}$$

substituting in the differential equation, we have

$$18e^{3x}=\dfrac{1}{2e^{3x}}×36e^{6x}\to18e^{3x}=18e^{3x}\Rightarrow$$  true

For option (D)

$$y=ln2x\Rightarrow\,\dfrac{dy}{dx}=\dfrac{1}{2x}×2=\dfrac{1}{x}$$ & $$\dfrac{d^2y}{dx^2}=\dfrac{-1}{x^2}$$

substituting in the differential equation, we have

$$\dfrac{-1}{x^2}=\dfrac{1}{ln2x}×\dfrac{1}{x^2}\to$$not true

$$\therefore$$ Option (C) is correct.

### Which of the following is a solution to the differential equation $$\dfrac{d^2y}{dx^2}=\dfrac{1}{y}\left(\dfrac{dy}{dx}\right)^2$$?

A

$$f(x)=y=sin^2x$$

.

B

$$f(x)=y=cos^2x$$

C

$$f(x)=y=2e^{3x}$$

D

$$f(x)=y=ln2x$$

Option C is Correct

# Finding the Value of a Parameter of Differential Equation for the Given Solution

•  A function $$f$$ is called solution to a differential equation if the equation is satisfied when $$y=f(x)$$ and its derivative are substituted in the equation.
• Suppose we desire to know the value of $$\alpha$$ for which $$y=e^{\alpha x}$$ is a solution to the differential equation.

$$y''+2y'+y=0$$

Find $$y'=\alpha e^{\alpha x}$$ and $$y''=\alpha ^2e^{\alpha x}$$ and put in the equation.

$$\alpha^2 e^{\alpha x}+2\alpha e^{\alpha x}+e^{\alpha x}=0$$

$$\Rightarrow \,e^{\alpha x}(\alpha^2+2\alpha+1)=0$$

$$\Rightarrow\,e^{\alpha x}(\alpha+1)^2=0$$

$$\Rightarrow\,\alpha=–1$$

#### If $$y=e^{\alpha x}$$ is a solution to the differential equation $$y''+5y'+6y=0$$. Then the possible value(s) of constant  $$'\alpha'$$ are

A $$\alpha=-2,\,\alpha=-3$$

B $$\alpha=-5,\,\alpha=\dfrac{1}{2}$$

C $$\alpha=1,\,\alpha=6$$

D $$\alpha=4,\,\alpha=1$$

×

$$y=f(x)$$ is a solution to a differential equation if the equation is satisfied when $$y=f(x)$$ and its derivatives are substituted in the equation.

In this case :

$$y=e^{\alpha x}\Rightarrow\,\dfrac{dy}{dx}=y'=\alpha e^{\alpha x}\Rightarrow\,\dfrac{d^2y}{dx^2}=y''=\alpha^2e^{\alpha x}$$

Putting the above in the differential equation

$$\alpha^2e^{\alpha x}+5\alpha e^{\alpha x}+6 e^{\alpha x}=0$$

$$\Rightarrow\,e^{\alpha x} (\alpha^2+5\alpha+6)=0$$  (we know $$e^{\alpha x}\ne\,0$$ for any $$x$$).

$$\therefore\,\alpha^2+5\alpha+6=0\Rightarrow\,(\alpha+2)(\alpha+3)=0\Rightarrow\,\alpha=-2,\,\alpha=-3$$

### If $$y=e^{\alpha x}$$ is a solution to the differential equation $$y''+5y'+6y=0$$. Then the possible value(s) of constant  $$'\alpha'$$ are

A

$$\alpha=-2,\,\alpha=-3$$

.

B

$$\alpha=-5,\,\alpha=\dfrac{1}{2}$$

C

$$\alpha=1,\,\alpha=6$$

D

$$\alpha=4,\,\alpha=1$$

Option A is Correct