第12类RD Sharma解–第22章微分方程–练习22.4

问题1.对于以下每个初始值问题，请验证附带函数是否为解：x(dy / dx)= 1，y(1)= 0

函数：y = logx

解决方案：

We have,

y = logx          -(1)

On differentiating eq(1) w.r.t x,

dy/dx = (1/x)

x(dy/dx) = 1

Thus, y = logx satisfy the given differential equation.

If x = 1, y = log(1) = 0 

So, y(1) = 0

问题2.对于以下每个初始值问题，请验证附带函数是否为解：(dy / dx)= y，y(0)= 0

函数：y = e ^x

解决方案：

We have,

 y = e^x          -(1)

On differentiating eq(1) w.r.t x

dy/dx = e^x 

(dy/dx) = y

Thus, y = e^x satisfy the given differential equation.

If x = 0, y = e⁰ = 1 

So, y(0) = 1

问题3.对于以下每个初始值问题，请验证附带函数是否为解：(d ² y / dx ² )+ y = 0，y(0)= 0，y’(0)= 1

函数：y = sinx

解决方案：

We have,

y = sinx           -(1)

On differentiating eq(1) w.r.t x,

(dy/dx) = cosx          -(2)

Again, differentiating eq(2) w.r.t x,

d²y/dx² = -sinx

d²y/dx² + sinx = 0

Thus, y = sinx satisfy the given differential equation.

If x = 0, y(0) = sin(0) = 0 

y'(0) = cos(0) = 1

问题4.对于以下每个初始值问题，请验证附带函数是否为解：d ² y / dx ² –(dy / dx)= 0，y(0)= 2，y’(0)= 1

函数：y = e ^x + 1

解决方案：

We have,

y = e^x + 1           -(1)

On differentiating eq(1) w.r.t x,

(dy/dx) = e^x           -(2)

Again differentiating eq(2) w.r.t x,

d²y/dx² = e^x         

d²y/dx² – e^x = 0        

d²y/dx² – (dy/dx) = 0

Thus, y = e^x + 1 satisfy the given differential equation.

If x = 0, y(0) = e⁰ + 1, y(0) = 1 + 1 = 2

y'(0) = e⁰ = 1

问题5.对于以下每个初始值问题，请验证附带函数是否为解：(dy / dx)+ y = 2

函数：y = e ^-x + 2

解决方案：

We have,

 y = e^-x + 2           -(1)

On differentiating eq(i) w.r.t x,

(dy/dx) = -e^-x

(dy/dx) + e^-x = 0

(dy/dx) + (y – 2) = 0

(dy/dx) + y = 2

Thus, y = e^-x + 2 satisfy the given differential equation.

If x = 0, y(0) = e^-0 + 2 = 1 + 2 = 3

问题6.对于以下每个初始值问题，请验证附带函数是否为解：(d ² y / dx ² )+ y = 0，y(0)= 1，y’(0)= 1

函数：y = sinx + cosx

解决方案：

We have,

 y = sinx + cosx           -(1)

On differentiating eq(i) w.r.t x,

dy/dx = cosx – sinx           -(2)

Again differentiating eq(ii) w.r.t x,

d²y/dx² = -sinx – cosx

d²y/dx² = -(sinx + cosx)

(d²y/dx²) + y = 0 

Thus, y = sinx + cosx satisfy the given differential equation.

If x = 0, y(0) = sin0 + cos0 = 1

y'(0) = cos0 – sin0 = 1

问题7.对于以下每个初始值问题，请验证附带函数是否为解：(d ² y / dx ² )– y = 0，y(0)= 2，y’(0)= 0

函数：y = e ^x + e ^-x

解决方案：

We have,

 y = e^x + e^-x           -(1)

On differentiating eq(i) w.r.t x,

dy/dx = e^x – e^-x           -(2)

Again differentiating eq(2) w.r.t x,

d²y/dx² = e^x + e^-x

d²y/dx² = y

d²y/dx² – y = 0

Thus, y = e^x + e^-x satisfy the given differential equation.

If x = 0, y(0) = e⁰ + e^-0 = 1 + 1 = 2

y'(0) = e⁰ – e^-0 = 0

问题8.对于以下每个初始值问题，请验证附带函数是否为解：(d ² y / dx ² )– 3(dy / dx)+ 2y = 0，y(0)= 2，y’( 0)= 3

函数：y = e ^x + e ^2x

解决方案：

We have,

y = e^x + e^2x           -(1)

On differentiating eq(1) w.r.t x,

dy/dx = e^x + 2e^2x           -(2)

Again differentiating equation(2) w.r.t x,

d²y/dx² = e^x + 4e^2x

d²y/dx² = 3(e^x + 2e^2x) – 2(e^x + e^2x)

(d²y/dx²) = 3(dy/dx) – 2y

(d²y/dx²) – 3(dy/dx) + 2y = 0

Thus, y = e^x + e^2x satisfy the given differential equation.

If x = 0, y(0) = e⁰ + e⁰ = 1 + 1 = 2

y'(0) = e⁰ + 2e⁰ = 1 + 2 = 3

问题9.对于以下每个初始值问题，请验证附带函数是否为解：(d ² y / dx ² )– 2(dy / dx)+ y = 0，y(0)= 1，y’( 0)= 2

函数：y = xe ^x + e ^x

解决方案：

We have,

y = xe^x + e^x            -(1)

On differentiating eq(1) w.r.t x,

dy/dx = xe^x + e^x + e^x  

dy/dx = xe^x + 2e^x           -(2)

Again differentiating eq(2) w.r.t x,

d²y/dx² = xe^x + e^x + 2e^x             

d²y/dx² = xe^x + e^x + 2e^x + xe^x + e^x – xe^x – e^x    

d²y/dx² = 2(xe^x + e^x) – (xe^x + e^x)

(d²y/dx²) = 2(dy/dx) – y

(d²y/dx²) – 2(dy/dx) + y = 0

Thus, y = xe^x + e^x satisfy the given differential equation.

If x = 0, y(0) = 0e⁰ + e⁰ = 1

y'(0) = 0e⁰ + 2e⁰ = 2