Solve partial u partial t =D partial^ 2 u partial x^ 2 + partial u partial x + lambda u equation is | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac { \partial u } { \partial t } = D \frac { \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial u } { \partial x } + \partial u } { \partial u u + \partial n }$$

Solve for D (complex solution)

$\left\{\begin{matrix}D=0\text{, }&n\neq -u^{2}\text{ and }∂\neq 0\\D\in \mathrm{C}\text{, }&u=0\text{ and }∂\neq 0\text{ and }n\neq 0\end{matrix}\right.$

Steps for Solving Linear Equation

Multiply both sides of the equation by $∂\left(n+u^{2}\right)$.

$$∂\left(n+u^{2}\right)\frac{\mathrm{d}(u)}{\mathrm{d}t}=D\left(\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u\right)$$

Use the distributive property to multiply $∂$ by $n+u^{2}$.

$$\left(∂n+∂u^{2}\right)\frac{\mathrm{d}(u)}{\mathrm{d}t}=D\left(\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u\right)$$

Use the distributive property to multiply $∂n+∂u^{2}$ by $\frac{\mathrm{d}(u)}{\mathrm{d}t}$.

$$∂n\frac{\mathrm{d}(u)}{\mathrm{d}t}+∂u^{2}\frac{\mathrm{d}(u)}{\mathrm{d}t}=D\left(\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u\right)$$

Use the distributive property to multiply $D$ by $\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u$.

$$∂n\frac{\mathrm{d}(u)}{\mathrm{d}t}+∂u^{2}\frac{\mathrm{d}(u)}{\mathrm{d}t}=D\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+D\frac{\mathrm{d}(u)}{\mathrm{d}x}+D∂u$$

Swap sides so that all variable terms are on the left hand side.

$$D\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+D\frac{\mathrm{d}(u)}{\mathrm{d}x}+D∂u=∂n\frac{\mathrm{d}(u)}{\mathrm{d}t}+∂u^{2}\frac{\mathrm{d}(u)}{\mathrm{d}t}$$

Combine all terms containing $D$.

$$\left(\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u\right)D=∂n\frac{\mathrm{d}(u)}{\mathrm{d}t}+∂u^{2}\frac{\mathrm{d}(u)}{\mathrm{d}t}$$

The equation is in standard form.

$$u∂D=0$$

Divide $0$ by $∂u$.

$$D=0$$

Solve for D

$\left\{\begin{matrix}D=0\text{, }&n\neq -u^{2}\text{ and }∂\neq 0\\D\in \mathrm{R}\text{, }&u=0\text{ and }∂\neq 0\text{ and }n\neq 0\end{matrix}\right.$

Steps for Solving Linear Equation

Multiply both sides of the equation by $∂\left(n+u^{2}\right)$.

$$∂\left(n+u^{2}\right)\frac{\mathrm{d}(u)}{\mathrm{d}t}=D\left(\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u\right)$$

Use the distributive property to multiply $∂$ by $n+u^{2}$.

$$\left(∂n+∂u^{2}\right)\frac{\mathrm{d}(u)}{\mathrm{d}t}=D\left(\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u\right)$$

Use the distributive property to multiply $∂n+∂u^{2}$ by $\frac{\mathrm{d}(u)}{\mathrm{d}t}$.

$$∂n\frac{\mathrm{d}(u)}{\mathrm{d}t}+∂u^{2}\frac{\mathrm{d}(u)}{\mathrm{d}t}=D\left(\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u\right)$$

Use the distributive property to multiply $D$ by $\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u$.

$$∂n\frac{\mathrm{d}(u)}{\mathrm{d}t}+∂u^{2}\frac{\mathrm{d}(u)}{\mathrm{d}t}=D\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+D\frac{\mathrm{d}(u)}{\mathrm{d}x}+D∂u$$

Swap sides so that all variable terms are on the left hand side.

$$D\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+D\frac{\mathrm{d}(u)}{\mathrm{d}x}+D∂u=∂n\frac{\mathrm{d}(u)}{\mathrm{d}t}+∂u^{2}\frac{\mathrm{d}(u)}{\mathrm{d}t}$$

Combine all terms containing $D$.

$$\left(\frac{\mathrm{d}(u)}{\mathrm{d}x^{2}}+\frac{\mathrm{d}(u)}{\mathrm{d}x}+∂u\right)D=∂n\frac{\mathrm{d}(u)}{\mathrm{d}t}+∂u^{2}\frac{\mathrm{d}(u)}{\mathrm{d}t}$$

The equation is in standard form.

$$u∂D=0$$

Divide $0$ by $∂u$.

$$D=0$$

Solve for n

$n\neq -u^{2}$
$\left(u=0\text{ or }D=0\right)\text{ and }∂\neq 0$