مسئله مقدار اولیه

${\displaystyle \frac{dy}{dt}=0.85y}$

${\displaystyle \frac{dy}{y}=0.85dt}$

${\displaystyle \ln |y|=0.85t+B}$

${\displaystyle |y|=e^B{e}^{0.85t}}$

${\displaystyle y=C{e}^{0.85t}}$

${\displaystyle 19=C{e}^{0.85\ast 0}}$

${\displaystyle C=19}$

${\displaystyle y^{\prime }+3y=6t+5,y(0)=3}$

${\displaystyle y(t)=2e^{-3t}+2t+1.}$

${\displaystyle y^{″}+A(x)y^{\prime }+B(x)y=F(x)}$

${\displaystyle y(a)=q_0,y^{\prime }(a)=q_1}$

${\displaystyle y^{\prime }(x)-q_1=-A(x)y(x)-{\int }_a^x[B(x^{\prime })-A^{\prime }(x^{\prime })]y(x^{\prime })d{x}^{\prime }+{\int }_a^{x}F(x^{\prime })d{x}^{\prime }+A(a)q_0}$

${\displaystyle y(x)-q_0=-{\int }_a^{x}A(x^{\prime })y(x^{\prime })d{x}^{\prime }-{\int }_a^x{\int }_a^{x_1}[B(x^{\prime })-A^{\prime }(x^{\prime })]y(x^{\prime })d{x}^{\prime }d{x}_1+{\int }_a^x{\int }_a^{x_1}F(x^{\prime })d{x}^{\prime }d{x}_1+[A(a)q_0+q_1](x-a)}$

${\displaystyle {\int }_a^x{\int }_a^{x_1}F(x^{\prime })d{x}^{\prime }d{x}_1={\int }_a^x(x-t)F(t)dt}$

${\displaystyle y(x)=q_0+[A(a)q_0+q_1](x-a)+{\int }_a^x(x-t)F(t)dt-{\int }_a^x\{A(t)+(x-t)[B(t)-A^{\prime }(t)]\}y(t)dt}$

${\displaystyle K(x,t)=-\{A(t)+(x-t)[B(t)-A^{\prime }(t)]\}}$

${\displaystyle f(x)={\int }_a^x(x-t)F(t)dt+[A(a)q_0+q_1](x-a)+q_0}$

${\displaystyle y(x)=f(x)+{\int }_a^{x}K(x,t)y(t)dt}$

${\displaystyle \frac{d^{n}y}{d{x}^n}+A_1(x)\frac{d^{n-1}y}{d{x}^{n-1}}+...+A_{n-1}\frac{dy}{dx}+A_n(x)y=F(x)}$

${\displaystyle y(a)=q_0,y^{\prime }(a)=q_1,...,y^{(n-1)}(a)=q_{n-1}}$

${\displaystyle \frac{d^{n}y}{d{x}^n}=g(x)}$

${\displaystyle \frac{d^{n-1}y}{d{x}^{n-1}}={\int }_a^{x}g(t)dt+q_{n-1}}$

${\displaystyle \frac{d^{n-2}y}{d{x}^{n-2}}={\int }_a^x(x-t)g(t)dt+(x-a)q_{n-1}+q_{n-2}}$

${\displaystyle \frac{dy}{dx}={\int }_a^x\frac{(x-t{)}^{n-2}}{(n-2)!}g(t)dt+\frac{(x-a{)}^{n-2}}{(n-2)!}{q}_{n-1}+\frac{(x-a{)}^{n-3}}{(n-3)!}{q}_{n-2}+...+(x-a)q_2+q_1}$

${\displaystyle y={\int }_a^x\frac{(x-t{)}^{n-1}}{(n-1)!}g(t)dt+\frac{(x-a{)}^{n-1}}{(n-1)!}{q}_{n-1}+\frac{(x-a{)}^{n-2}}{(n-2)!}{q}_{n-2}+...+(x-a)q_1+q_0}$

${\displaystyle K(x,t)=-{\mathrm{\Sigma }}_{k=1}^n{A}_k(x)\frac{(x-t{)}^{k-1}}{(k-1)!}}$

${\displaystyle f(x)=F(x)-q_{n-1}{A}_1(x)-[(x-a)q_{n-1}+q_{n-2}]A_2(x)-...-\{[\frac{(x-a{)}^{n-1}}{(n-1)!}]q_{n-1}+...+(x-a)q_1+q_0\}{A}_n(x)}$

${\displaystyle g(x)=f(x)+{\int }_a^{x}K(x,t)g(t)dt}$