Answers:
1. P=3
Step by step:
P-5=-2 -move the constant to the right hand side and change its sign
P=-2+5 - calculate the sum
P=3
2. W=-2.8
Step by step:
W+13.2=10.4 - move the constant tot he right hand side and change its sign
W=10.4-13.3 - calculate the difference
W=-2.8
3. X=2/3
Step by step:
X-5/6=-1/6 - move the constant to the rig hand side and change its sign
X=-1/6+5/6 - calculate the sum
X=2/3
Answer:
190
Step-by-step explanation:
X=(b/2m)*y-e is the answer I believe.
Answer:
Therefore the particular solution of the given differential equation is
![Y(x)=\frac{1}{6} x^2 ln x](https://tex.z-dn.net/?f=Y%28x%29%3D%5Cfrac%7B1%7D%7B6%7D%20x%5E2%20ln%20x)
Step-by-step explanation:
The given ordinary differential equation is
![x^2y"-3xy' +4y=11x^2](https://tex.z-dn.net/?f=x%5E2y%22-3xy%27%20%2B4y%3D11x%5E2)
If y₁(x) =x² is a solution of ODE then it will be satisfy the ODE
y₁'(x)= 2x and y₁"(x)=2
Putting the value of y₁'(x) and y₁"(x) in x²y"-3xy'+4y=0 we get
x².2-3x.2x+4x²= 2x²-6x²+4x²=0
Therefore y₁(x) is a solution of the given differential equation.
Again,
y₂(x) =(x²ln x ) is a solution of ODE then it will be satisfy the ODE.
![y'_2=2x ln x+ x^2\times\frac{1}{x}](https://tex.z-dn.net/?f=y%27_2%3D2x%20ln%20x%2B%20x%5E2%5Ctimes%5Cfrac%7B1%7D%7Bx%7D)
![= 2x ln x+x](https://tex.z-dn.net/?f=%3D%202x%20ln%20x%2Bx)
= 3+2 ln x
Putting the value of y₂'(x) and y₂"(x) in x²y"-3xy'+4y=0 we get
=0
Therefore y₂(x) is a solution of the given differential equation.
The wronskian of y₁(x) and y₂(x) is
![W(y_1,y_2)(x)=\left|\begin{array}{cc}y_1&y_2\\y'_1&y'_2\end{array}\right|](https://tex.z-dn.net/?f=W%28y_1%2Cy_2%29%28x%29%3D%5Cleft%7C%5Cbegin%7Barray%7D%7Bcc%7Dy_1%26y_2%5C%5Cy%27_1%26y%27_2%5Cend%7Barray%7D%5Cright%7C)
![=\left|\begin{array}{cc}x^2&x^2 ln x\\2x&2x ln x+x\end{array}\right|](https://tex.z-dn.net/?f=%3D%5Cleft%7C%5Cbegin%7Barray%7D%7Bcc%7Dx%5E2%26x%5E2%20ln%20x%5C%5C2x%262x%20ln%20x%2Bx%5Cend%7Barray%7D%5Cright%7C)
=x²(2x ln x+x)-x²ln x(2x)
=2x³ ln x +x³ - 2x³ln x
=x³≠0
Here ![g(x)=\frac{y_2(x)}{y_2(x)} = ln x](https://tex.z-dn.net/?f=g%28x%29%3D%5Cfrac%7By_2%28x%29%7D%7By_2%28x%29%7D%20%3D%20ln%20x)
The particular solution is
![Y(x)=- y_1(x)\int\frac{y_2(x).g(x)}{W(y_1,y_2)(x)}dx + y_2(x)\int\frac{y_1(x).g(x)}{W(y_1,y_2)(x)}dx](https://tex.z-dn.net/?f=Y%28x%29%3D-%20y_1%28x%29%5Cint%5Cfrac%7By_2%28x%29.g%28x%29%7D%7BW%28y_1%2Cy_2%29%28x%29%7Ddx%20%2B%20y_2%28x%29%5Cint%5Cfrac%7By_1%28x%29.g%28x%29%7D%7BW%28y_1%2Cy_2%29%28x%29%7Ddx)
![=-x^2\int \frac{(lnx)^2}{x} dx+x^2 ln x\int \frac{ln x}{x} dx](https://tex.z-dn.net/?f=%3D-x%5E2%5Cint%20%5Cfrac%7B%28lnx%29%5E2%7D%7Bx%7D%20dx%2Bx%5E2%20ln%20x%5Cint%20%5Cfrac%7Bln%20x%7D%7Bx%7D%20dx)
Let ln x =u ![\Rightarrow \frac{1}{x} dx=du](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7B1%7D%7Bx%7D%20dx%3Ddu)
![=-x^2 \frac{u^3}{3} +x^2 ln x \frac{u^2}{2}](https://tex.z-dn.net/?f=%3D-x%5E2%20%5Cfrac%7Bu%5E3%7D%7B3%7D%20%2Bx%5E2%20ln%20x%20%5Cfrac%7Bu%5E2%7D%7B2%7D)
![=-\frac{x^2(lnx)^3}{3} +\frac{x^2(lnx)^3}{2}](https://tex.z-dn.net/?f=%3D-%5Cfrac%7Bx%5E2%28lnx%29%5E3%7D%7B3%7D%20%2B%5Cfrac%7Bx%5E2%28lnx%29%5E3%7D%7B2%7D)
![=\frac{1}{6}x^2 (ln x)^3](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B6%7Dx%5E2%20%28ln%20x%29%5E3)
Therefore the particular solution of the given differential equation is
![Y(x)=\frac{1}{6} x^2 ln x](https://tex.z-dn.net/?f=Y%28x%29%3D%5Cfrac%7B1%7D%7B6%7D%20x%5E2%20ln%20x)